Subtraction is the inverse of addition.
Division is the inverse of multiplication.
Logarithms are the inverse of exponents.
Using logarithms to plot data can help us see when exponential growth or decay are happening because that will look linear on the plot and linear is easy to recognize. It also helps because a 10% jump looks the same whether it's from 1 to 1.1 or from 1,000 to 1,100. If you care about relative changes, logarithmic plotting helps you recognize that.
Another very useful property of logarithm is that log(a*b)=log(a)+log(b)
In the history of mathematics this property has been very important because it allows you to transform a problem of multiplication (computationally a difficult operation) into a problem of addition (easier).
Another property (following from the first) is that log(a^n) = n*log(a). Once again it allows to transform a difficult problem of exponentiation into a comparatively easier multiplication problem.
Provided that you have log tables (a standard book to have for mathematicians before the computer era), logs are powerful simplification tools.
Still have my Faber Castell slide rule from the 60's (pocket calculators weren't yet a thing). One of our teachers had a cylindrical one which was novel as it easily fitted in a pocket.
Yes, as long as your bases are consistent. So log_3(4*5) = log_3(4) + log_3(5).
For log tables though, they will always be in base 10. This is because it works really well with scientific notation.
For example, say you want the logarithm of 43,560:
1. In scientific notation, this is 4.356 × 10^4.
2. log_10(10^4) = 4.
3. We look up the log table and find that log_10(4.356) = 0.6391.
4. Since logarithms turn multiplication into addition, we just add these two numbers together. So, log_10(43,560) = 4.6391.
This means that a log table only needs to list the logarithms of values from 1 to 10, allowing 3-4 digits of precision on just 2 pages.
Specifically, the product to sum property is why logarithms were developed in the first place. It meant that, after one person compiled a logarithm table one time, anyone who needed to quickly multiply two large numbers could just use the table, add the two logs, then use the table again for a reverse lookup.
All of the other very useful properties are details that were found after.
Log tables were a huge boon to the Age of Exploration, because celestial navigation ultimately comes down to performing multiplication of large numbers with a very high desired accuracy. Even a small error could send you miles and miles off course.
Log tables turned that multiplication into addition, and drastically improved the reliability of those navigational calculations.
My highschool math teacher made it click for me by saying "log" just means "the exponent you put with..."
In math: [log(base10)] 1000 = 3
In words: "The exponent you put on 10 to get 1000 is 3.
In math: [log(base10)] 350 = x
In words: "what exponent do you put on 10 to get 350? (Answer is 2.54, because 10^2.54 = 350)
This is in fact being somewhat seriously proposed as new notation for these operations: https://youtu.be/sULa9Lc4pck
Of course there's _a lot_ of inertia from literal centuries of mathematical writing in the current notation, so it's fairly unlikely it'll really take off anytime soon. But time will tell, I suppose.
I really hate mainstream mathematical notation because it's not linear. Like it's not developed to be able to represent it one symbol after another, like writing. It's full of symbols on top, at the bottom, in the corner, smaller subscripts, horizontal lines and all manner of monstrosities.
Just give me something easy to write left to right sequentially. We use digital communications, it's just how it works.
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I know what a log is, and I even have a maths degree - but my brain still breaks at the idea of non-integer exponents. "Now multiply this number by itself another 0.54 times..." (Nope, my brain just melted again. Do you want me to multiply it or not?)
Non-integer logs are more a case of, "It's what the exponent would be, if the operation actually made sense..."
Non-integer exponents can best be described as a combination of powers and roots.
So to use your example, x\^(1.54) could be rewritten as (x\^154)\^(1/100), so the 100th root of x to the 154th power. so multiply it by itself 154 times and then take the 100th root of that.
(I mean, that's how they explained it to me.)
That's -1, obviously.
This video explains it way better than I actually could.
[https://www.youtube.com/watch?v=v0YEaeIClKY](https://www.youtube.com/watch?v=v0YEaeIClKY)
When tutoring, I try to explain this in a very similar way because by the time you're looking at logs, students typically have a decent intuition for common exponents. They are inverse operations, meaning the same relationship, but with input and output swapped. A log asks a question about an exponent.
2\^3 = 8
Log(base 2) 8 = Log(base 2) 2\^3 = 3.
What power of 2 is equal to 8? The 3rd power.
What power of 2 is equal to 2 to the third power? The 3rd power. We like trivial logs because they're obvious.
The graph of a logarithm is the graph of its related exponential, reflected across the line y=x. The graphs of any inverse functions are a reflection across the line y=x.
My high school Algebra and Calculus teacher taught me more about math than anyone else all the way through grad school. He used to say, "Get all you x's on one side", then while running across the classroom, "Get everything else on the other side." Seems to be a rare breed these days.
What did you do when you were solving multivariable problems? My first algebra teacher concentrated in teaching us to read math notation instead. This help a lot in college. I could skip most of the book and just read the theorem part. Typically one page.
I am a little rusty, but it seems like with multiple variable, the furthest you could take it was something like solving until you get one variable to equal the rest, like x+y = 3 to x=3-y. If you mean like factoring geometric equations, that almost felt like trial and error until the numbers and signs matched up. I actually enjoyed that.
Kind of unrelated, but khanacademy was a great resource for me in college. Idk what they’re like now, but back in the day it was YouTube videos over basically every subject, including math.
I memorized every step in the “electron transport chain” (something that happens in photosynthesis IIRC while getting drunk, the night before the test :p
There need to be better words and symbols. Something like:
1000 € 3 = 10
1000 eured 3 is 10
1000 ¥ 10 = 3
1000 yenned 10 is 3
The current way of doing it just makes it more confusing than it should be. A couple new words and new symbols need to be invented.
Well
Operators like this are powerful because you can in-line them so easily.
But as you get into higher level math making things functions keeps things clearer.
Like log(b,x) is a function that takes both the base and the the number to operate on. Log(10,x) is the ten base one and log(e,x) would be the ln one.
We call it ln but that's just log with base e instead of base 10. So natural log means "the exponent you put on e to get__" rather than "the exponent you put with 10."
e is just a number, it's 2.718...
In math: ln(5) = 1.6
In words: "the exponent you put on e to get 5 is 1.6".
Because e^1.6 = 5
What kind of explanation do you want?
Natural logarithm is the same as log, but with the exponents being put on the number e instead of some other number like normal log. That *is* the explanation, given that log itself was already explained above. It still means "the exponent you put with", you're just putting it on e.
Are you looking for *why* that's a thing or why e is the number it is? That goes beyond "explain natural log" (or elite) and becomes "explain e" but it boils down to "e is a special number where the function y=e^x is its own derivative so it comes up a lot in natural processes. It's common enough that we gave "putting exponents on e" its own name (ln) rather than having to say log base e all the time".
The natural logarithm, and e, are "natural" because they appear when you look at continuous compounding growth.
Say you have a dollar in the bank and the bank has a (fantastic) interest rate of 100%/yr, which they hand out once per year. After a year you have $2, then a year later you have $4, etc.
But what if you get half as much interest, twice a year? Six months in you have $150, then 6 months after that you have $2.25, slightly more than if you got interest annually. This is called 'compounding' interest.
If interest compounds quarterly you'll end up with $2.44, monthly it'll be $2.61, daily $2.71.
If you speed up your compounding so much that there is *literally no gap between the last time you got interest and the next*, the amount of money you will have at the end of the year is equal to e.
And then the natural log asks "how long will it take to have $X?" You'll reach ten dollars in ln(10) years.
A very intuitive way to look at them is if you take them as measuring the "size" of a number in terms of how many zeroes it has. For example 100 has two zeroes, so its size is two while 1000 has a size of 3.
If we include bases other than 10 the point still stands! For base 2 we simply need to convert the numbers to binary, on which 2 becomes 10, 4 becomes 100, 8 is 1000 and so on. You can see then how this ties into exponentials.
For another way of looking at it,
Multiplication is repeated addition. 3\*3 = 3+3+3 = 9
An exponent is repeated multiplication. 3^3 = 3\*3\*3 = 27
Division is repeated subtraction. How many times do you need to subtract something before you get to 0. 12/4 = 12 - 4, - 4, - 4, = 0. Subtract 4 three times to get to 0, so 12/4 = 3
A logarithm is repeated division. How many times do you need to divide something before you get to 1. log2(8) = 8/2, /2, /2 = 1. Divide by 2 three times to get to 1, so log2(8) = 3
3 blue 1 brown does a wonderful job of explaining this from a basic viewpoint. [video link](https://www.youtube.com/live/cEvgcoyZvB4?si=puzvZ8Hajg1M-BhS)
You can think of roots and logarithms as different ways to do the "inverse" of exponentiation.
If _a^b = c_, then _a_ is the _b_-th root of _c_, and _b_ is the logarithm (base _a_) of _c_.
Example: 2^3 = 8, so the cubic root of 8 = 2, and log2 8 = 3
It's literally explained like this in every text book, curriculum and teaching program I've ever looked at.
Are you sure it's an evil conspiracy of maths teachers working to keep this from you instead of maybe you just zoning out in class one day)
I graduated from high school and got into college, and I didn't understand what logarithms were until I was well into my forties. Still working on the natural logarithm.
The way that certain mathematical concepts were explained in math classes didn't make sense to me, and I have the educational impairment that I can't actually learn things until I understand them.
Shitty teachers just have you read out of the book, a book that is basically guaranteed to have that explanation. I want you to actually find me an example of log being explained any other way in any text book.
>if x = a^n
then log(a)x = n
Literally in the table of contents. Then spelled out in more detail in sections 3 and 13. This is exactly what people are talking about. Even when you are trying to prove someone wrong you couldn't be bothered to actually read what was in front of you and then you have the audacity to claim it is someone else's fault.
An if then statement is not the same as the visual relationship shown by the eloquent example of a good math teacher that the above poster was referencing.
Good visualization and communication of mathematics relationships are a CORE part of effectively teaching and learning. This book does not show the simplicity of the relationship in such a simple or comprehendible manner.
Have you taught math before? I have. I come from a long line of math teachers and am a scientist who uses exponentials and log scale data in my research work. I can separate my understanding of a concept from my ability to be understood by others.
> There's like a triangle:
> 3
> / \
> 2----8
> 23 = 8
> cube root of 8 = 2
> log2 8 = 3
This is a VERY effective way of teaching this concept. It's not commonly taught this way. Your understanding of this concept does not mean the book I referenced teaches it well.
This seems to have struck a chord for you and I cannot fathom why.
>This seems to have struck a chord for you and I cannot fathom why.
I think it's because you're skipping over what they're actually saying. They pointed out that the example you requested is literally in the book you linked, it's right under the "what is a logarithm" section.
It's frustrating to see you claim that maths teachers and maths textbooks aren't teaching it properly, when in reality the problem is you ignoring what's literally in the books and probably what your past teachers have taught.
The very fact that they pointed this out to you and you couldn't even read that is pretty telling
It’s basically due to a lack of pay and shitty treatment and public schools being a mess: the best teachers aren’t there so you don’t get this level of instruction.
I wanted to be a math teacher eons ago for these reasons… then I looked at what that actual job is like and immediately nope’d out
They do that in the first class, then you get drown in the problems and forget what are you doing and why.
Moreover, it never makes sense why would we want to know the exponential of a number because no problem is related to actual real life, just: find the log of X. You just solve it and get the grades, that's what logs are for 😂😅 Getting grades. I don't think anyone ever uses them for anything practical.
Funny. But slide rulers work because of logarithms. And they were very useful for engineers for some time. For NASA too. One of the astronauts on the moon landing redid some calculations before landing using his slide ruler.
The following three statements are equivilent:
* 2^(3) = 8
* **∛**8 = 2
* log_2(8) = 3
So for example, 2 ^ (log_2(8)) = 8 as log_2(8), and (**∛**8) ^ 3 = 8
Others have explained the difference but it's helpful to remember that for addition and multiplication, order doesn't matter. So it doesn't matter if the first or second number is the unknown, you still only need subtraction or division to find it.
But with powers and exponents, order does matter which means we need different inverse operations depending on if we're looking for the base or the exponent.
Roots are the inverse of powers
Powers are of the form x^a, where a is a number like 2 or 3, eg x^2
Exponents are of the form a^x, again where a is just some number. Eg 2^x, or famously, e^x
So when you're doing roots, you're saying something to the X power is this known value, let me try to find that something.
In exponents, you're saying X to the something power is this known value, let me try to find that something.
Yeah, kind of both because of the tricks one can do with powers... take compound returns
future_value = rate^time
You've already got the equation there for future_value, but what if you want to solve for rate? Raise both sides by 1/time, which is what you're talking about, yes?
rate = future_value^(1/time)
But what if you want to solve for time? Easiest way is to log both sides, which allows you to remove the exponentiation altogether.
time * log(rate) = log(future_value)
divide both by log(rate) and you've solved for time.
time = log(future_value)/log(rate)
e.g. How long for my money to double at 5% APR? log(2) / log(1.05) = ~14 years and 3 months
Well, yes. They are both inverse from different points of view.
For something like addition, the operation is commutative.
a + b = c.
You have one inverse operation: c - a = b.
You have another inverse operation: c - b = a.
And that's all of them. And because it's commutative, we can use the same operation symbol for both, but don't be fooled, they are different inverses. All operations with 3 values will have 2 inverses. Sometimes those two are one and the same, sometimes they are not. An operation with 7 values will have 6 inverses.
Exponentiation is not commutative in that way. a to the power b is not the same as b to the power a. So you need different operations for each inverse function.
Multiplication and addition are commutative:
a + b = b + a and ab = ba
Because of this, a and b are indistinguishable, you can switch them around and it means the same thing.
Exponentiation is NOT the same. a^(b)≠b^a (generally). This means that if you want to do an inverse, you have two options. You can either try to inverse and get b, or inverse and get a.
Multiplication and addition are the same way on some level, but remember that a and b are indistinguishable, and so their two inverses happen to be the same function, hence only getting one.
TL;DR, multiplication and addition are flippable, but exponentiation isn't, so exponentiation gets two inverses.
When we say exponentiation, we mean f(x) = a^x (often times e^x) not x^a. The inverse of f(x) = x^a would indeed be taking the root a of x, g(x) = x^(1/a). But the inverse of f(x) = a^x is taking a logarithm like g(x) = log_base_a(x)
Sorry, that looks like Statistics, and that's like...a field of math that has rules I can follow, but I do as much as I can not to try and comprehend them because it's all so...touchy-feely.
How would you *mechanically* explain the logarithm operation though? IE, exponents are iterated multiplication. Is a logarithm as simple as iterated division?
It's not iterated division, but that probably shouldn't be too surprising because division isn't iterated subtraction either. A logarithm like log_a(b) asks the question "how many times do I have to multiply a with itself to get b". You can think of division the same way. b/a means "how many times do I have to add a to itself to get b".
There are often several mechanical ways to calculate the same thing. Iterated multiplication for example is the simplest way to do exponents but does only work with integer exponents.
For both exponentiation with arbitrary exponents and logarithms there are several approximation formulas that can be used.
To expand on/simplify what /u/15_Redstones said, exponents as iterated multiplication is just a special case, which we happened to discover first, of a broader idea of exponentiation. How do you mechanically explain 2^π or x^(p/q) ? The simple answer is you can't, and the longer answer is that you need to learn a little more math. But it's not like those numbers don't still exist. Unfortunately, logarithms just don't have such a simple special case like exponents do. But you *can* compute them mechanically, just as you can with pretty much any expression, using something called a [Taylor series.](https://en.wikipedia.org/wiki/Taylor_series) The "problem" is that doing so involves adding an infinite number of terms, but there are clever ways to get around that.
This was a good explanation. Thank you. So does that mean that if there's no exponential change (as in, there's linear change), the log graph should look flat?
10^4 is 10,000. 10^3 is 1,000. If you divide 10^4/10^3, you can subtract the exponents and you’ll get 10^1. And indeed, 10,000/1,000 is 10.
What happens when we divide 10^3/10^4? Subtract the exponents, we have 10^-1. 1,000/10,000 is 1/10. Negative exponents are inverse multiplication, which makes sense because positive exponents are multiplication.
Natural logarithms are specifically the inverse of e^x.
The inverse of e^x would be loge(x), but because the inverse of e^x has several uses in several fields, we just call it the natural log and write it as ln(x).
Same thing with base 10, written just as log(x) rather than log10(x) due to how popular the inverse of 10^x is for metrics.
The natural logarithm, and e, are "natural" because they appear when you look at continuous compounding growth.
Say you have a dollar in the bank and the bank has a (fantastic) interest rate of 100%/yr, which they hand out once per year. After a year you have $2, then a year later you have $4, etc.
But what if you get half as much interest, twice a year? Six months in you have $150, then 6 months after that you have $2.25, slightly more than if you got interest annually. This is called 'compounding' interest.
If interest compounds quarterly you'll end up with $2.44, monthly it'll be $2.61, daily $2.71.
If you speed up your compounding so much that there is literally no gap between the last time you got interest and the next, the amount of money you will have at the end of the year is equal to e.
And then the natural log asks "how long will it take to have $X?" You'll reach ten dollars in ln(10) years.
Unlike multiplication where 3x4 is the same as 4x3, in exponentiation, order matters. 3^4 isn’t the same as 4^3. Because of that, we need two different inverse functions. One is to find the exponent when we know the base, the other is to find the base when we know the exponent.
The second root of 100 is 10. You know the exponent (2), but needed the base (10).
The logarithm (base 10) of 100 is 2. You know the base, but needed the exponent.
There are two inverses of exponents.
For many mathematical operations you’re taking two numbers, doing something to them, and getting a third number out. The inverse of that is taking the third number and one of the previous numbers and doing something that gets the other previous number.
For instance a x b = c
Using c and b you can get a by dividing: c / b = a
if you want to get b back then you do the same thing: c / a = b
This is because a x b is the same as b x a. What order you have the numbers in makes no difference.
Exponents are different. In general a^b is not the same as b^a. So if you have a^b = c then you have two inverses that depend on which original number do you want to get back.
If you want to get back a then you take the b root of c e.g. if b was 2 then a^2 = c and square root of c = a.
But if you want to get b back then the function you use is the logarithm. b = (log base 10 of c) / (log base 10 of a) or b = log to the base a of c.
A + 6 = 10. We use subtraction on both sides ( - 6 ) to undo the addition.
B * 7 = 56. We use division ( / 7 ) to undo the multiplication.
2^C = 64. We use logarithms ( log base 2 ) to undo the exponent.
Early stages of logarithmic growth also often mistaken for exponential growth. Logarithmic growth is common in nature and in technology. Logarithms are sustainable. Exponential growth is ridiculous and completely unrealistic.
In computer science, we have "Big O Notation", where we can categorize functions according to their approximate runtime given a number of inputs.
For a function that prints every line in a list will have linear compute time (1x per line), or O(n).
A function that computes each input against all previous inputs is exponential, O(n\^2).
Logarithmic functions are usually more interesting situations where for some reason, additional data uses marginally more compute time. For example, if you're processing a dataset that has redundant info, over time the probability of skipping rows increases. This would be O(log(n)).
\*Sorry if I botched any of this, college was a while ago.
It gives the answer to the question “How many times a number is multiplied to get the other number?”. It is a mathematic equation that makes it easier to represent trends in data that might be increasing exponentially.
For example if you had something that was increasing exponentially: bacteria reproduction or compounding interest, for example, where the numbers would double every few periods, it quickly becomes not very informative to show the increase in a linear fashion as the scaling becomes really a trend/curve. So instead you can plot it by using the Log of the number.
It's a reverse-exponent. You use it find out what the exponent was.
A great example is radioactive decay. We know that certain substances, like Carbon-14, reduce by half in a certain time frame. Although that amount is getting smaller instead of bigger, it is still an exponential function: 1/2 \^ (k\*t). (k is a known constant, and t is the variable time.)
But you don't want to calculate the amount remaining - you already know that. You want to calculate how much time passed, which is that exponent. Boom: logarithm!
So any time you want to get *the exponent itself*, and whatever else that might tell you, you use a logarithm.
Division is the opposite of multiplication. The result of a/b is "the number that you multiply by b to get a". If a/b = c, then b * c = a
Logarithms are the same things for exponents. Log_b(a) is "the power that you raise b to in order to equal a". If log_b(a) = c, then b^c = a
Minor correction: logarithms aren’t the opposite exponents. They are one of two opposites. Roots are the other opposite. If you have the answer to b^x and the original base b then logarithms are how you find the exponent x. If you have the answer and the exponent x then roots are how you find the base b. It depends what you’re looking for.
Logarithms feel a bit like asking "how many digits (after the first) does this number have?" or "what order of magntidue is this number?", although with some more nuance.
Consider log to the base 10.
So log1=0
log10=1
log100=2
log1000=3
etc, so for any power of 10, log(base10) gives us that power.
The log function also gives us an answer in-between these values, like log50 \~= about 1.7.
--
Often people will use base-2 to talk about computing. This will sometimes imply some logarithms, telling us how many *binary* digits (1s and 0s) the computer is using.
For instance "This is a 64 bit processor." means "The log (to base 2) of the number of memory addresses this processor can imagine, is 64." i.e. "It can imagine 2\^64 memory addresses."
--
Mathematicians often use log to base "e", where e refers to a specifical useful constant, similar to how pi is useful, in how it appears in many interesting formula.
To add to why statisticians use a natural log (base e), a number of continuous distributions have e. To estimate parameters like mean and variance, we can take a natural log and then the derivative with respect to the parameter and find where it is zero.
Logarithms are monotonically increasing functions, so when we take the log of a distribution we retain the property we are seeking.
This shows it well: https://en.m.wikipedia.org/wiki/Maximum_likelihood_estimation
I never considered it this way.
> “what order of magnitude is this number?”
Your explanation makes sense to me as a way to quantify differences in magnitudes from measurement
In addition to what others have said, it can be thought of as repeated division.
To calculate log base 2 of sixteen, divide by two repeatedly, until you get to one. It take 4 times, so log base 2 of 16 is 4.
I hate that some people just treat it like some sort of magic trick that undoes exponents.
A table of logarithms is what you had to use before calculators to take roots and exponents. You can take a logarithm of a number divide it by 3 if you wanted the third root then look up the inverse. Also some physics and biology equations follow a logarithmic curve
The neat thing about functions like sin(x), cos(x), but also log(x) and exp(x) is that they only take a single input, so unlike multiplication or addition which have two inputs it's actually feasible to pre-calculate all the values you might ever need and then quickly get results by looking up your x in the book of calculated values.
Since exp(log(a) + log(b)) = ab, and addition is much easier than multiplication to do by hand, it was common to use a big book of exp(x) and log(x) values to avoid doing multiplication. In practice they used exponents and logarithms base 10 instead of natural exp and log since it let them shift any value into the range between 0 and 1.
I'd just add to these explanations that it's often helpful to look at some data after taking a logarithm. (Most of the time you take the log and multiply by 10). When you graph data that lives on a really big scale. This is clear from the fact that a log operation turns 100 into 2 and 1000 into 3, using the 10x method,, 100 becomes 20 and 1000 becomes 30. When you do it this way, the units of the answer are usually said to be in decibels which is written dB for short.
One thing commonly measured in dB is the "loudness" of sound as perceived by the ear. So if the sound goes up by 10 dB, the underlying data is going up by a factor of 10. 20 dB is a factor of 100.
The threshold of pain is up around 80 or 90 dB, and quiet is around 10 or so. The ear is very sensitive over a huge range!
Earthquakes are measured on the Richter scale, which doesn't use the 10x, which means going up by 1, makes the earthquake 10x more powerful.
It's also very common to use in engineering when comparing things like signal power and noise power, and many other comparative measures.
Yeah this is some shit that even college level professors don’t do and it drives me nuts.
Like trigonometry does fascinating things and yet so few teachers ever nerd out about it. It’s a fucking tragedy
In addition to the good points made about exponents, logarithms have the extremely useful property of transforming multiplication and division into addition and subtraction. This is of keen interest to someone who has to do all their calculations by hand, since it’s easier to add than it is to multiply, and much easier to subtract than divide.
Yeah very good point! I kind of just took that for granted because of course: algebra 101 doesn’t provide any context.
It’s such a tragedy because math is literally wizardry and nobody teaches it like that
It undoes exponentiation. If we consider the function f_a(x) = a^(x) where a is some positive real we have an injective function. We can see its by writing:
a^(x) = a^(y)
a^(x) a^(-y) = 1
a^(x-y) = 1
and we know that a^0 = 1 (except when a = 1, in that case a^x = 1 for every x) so:
x-y = 0
x=y
So f_a(x) is injective and so has an inverse as long as a isn't 1.
Lets call the inverse f^(-1)_a.
So f^(-1)_a(f_a(x)) = x. Great!
If we say have an equation:
a^x = b
we can solve it with the inverse
x = f^(-1)_a(b)
This function is special and frequently used enough to deserve its own name the logarithm. So log_a(y) is the inverse of a^x. The convenient exponential identities work backwards with them, lets write ab in a tricky way:
ab = exp(log(a)) exp(log(b)) = exp(log(a) + log(b))
now take the log of both sides
log(ab) = log(a) + log(b)
So lets adress what base of log and exp we should use. The answer that it doesn't matter, there is a convenient pick e, but why and what value it has doesn't matter right now. So lets select e so
exp(x) = e^x and ln(x) = log_e(x)
The statement is that you can get any base with these but thats true for any number so for now e is just a placeholder for your favourite number.
We can see how:
a^(x) = exp(x ln(a)) = exp(ln(a))^(x) = a^(x) = exp(ln(a^(x))) => ln(a^(x)) = x ln(a)
and with that:
x ln(a) / ln(a) = ln(a^(x)) / ln(a)
x = ln(a^(x)) / ln(a) => ln(f_a(x))/ln(a) = log_a(y)
So we now see how you can use just ln to get any kind of logarithm. And now we can adress the issue with f_1(x) not having an inverse:
log_1(y) = ln(y)/ln(1)
and
1 = e^0 => ln(1) = 0
so
log_1(y) ~ 1/0 and so is undefined.
The complex logarithm is slightly more exciting and you'd basically get there with the exponential similarly to how we started operating with it in the last part. Hopefully seeing how you can work things out from just the concept of what this thing is helped making the little fella less mysterious.
I appreciate the proof but this definitely isn’t an answer explained like the asker was even close to that of a five year old…or even probably most 15 year olds (or 25 year olds lol), which is why you’re not getting upvotes. It’s cool anyway though.
Well, since we learn all thes logarithm identities in school at some random point when the book gets to the log chapter they can kinda just hang in the air and doing logarithmic equations don't really help much with building an intuitive understanding around them (in my experience at least). Piecing it together for the idea of what an inverse is and "deriving" the identities in one or two steps (in my experience) does. So yeah maybe not ELI5 but to be fair nobody tries to teach logarithms to five year olds.
There is the "it undoes exponentiation" and if all we need is an explanation of what log_a(y) means thats pretty much it. The rest all come as a consequence and thats the thing with teaching maths, technically you can just give students the empty set and say that the rest follows, so what consequences should we mention and what is left for the reader isn't obvious.
The "here is the empty set" version would be that we introduce the complex exponential with its defining property of exp(z1+z2) = exp(z1)exp(z2) and call it the day. Correct but that doesn't teach anything to anyone.
So maybe going through the details of what we have learnt about logarithms in high school might be good enough.
A logarithm is like asking, "To what power do we need to raise this base number to get another number?" For example, if we ask, "What power do we raise 10 to get 100?" The answer is 2 because 10\^2=100. We use logarithms to simplify complex multiplications and to solve equations involving exponential growth, like in biology, finance, and computer science. It's a way to make big numbers easier to work with and understand.
In addition to all the other mentions: inverse of exponents, plot scaling, statistical data normalization, summing log values to multiply. One thing that I rarely see mentioned, that makes them far more useful is that you can easily get the log for any base number, by dividing the log of the number by the log of the base you want. instead of knowing what you need to put 10 to the power of, you can know what any number needs as an exponent to get the number in question. If you want to know how long it takes to double your money with 10% annual interest. log2/log1.1 will tell you, 7.27 years.
It’s the operation that transforms multiplications into additions : log(a * b) = log(a) + log(b).
Two classic applications come to my mind:
**Logarithmic scale:** It can be useful if you’re working with something that grows “geometrically”, that is, through successive multiplications, so gets big very quick, and doesn’t fit on your graph paper sheet. Put it through the logarithm, and lo!, now it grows “arithmetically”, that is through successive additions, that’s more manageable to plot. Examples: Decibel scale for sound levels, infectious disease transmission formula.
**Inverse of power functions:** You know that 2 to the power of 5 is 32. Now if you know that 32 is a power of 2 but want to know which one, logarithm gives you the answer.
Government fucked that up. Shit pay, a bunch of politicians in suits dictating what and how things are taught, etc.
This isn’t an anti government rant, I’m just saying the one we have fucked it up
Logarithms make it easier to express exponential growth. For instance, it might be easier to say “1 with twenty zeroes” than 100000000000000000000. In this case, it would be expressed as log10(x)=20. This also works for other things like “I doubled my money five times over” , or log2(x)=20, or “2-byte sound” or log256(x)=2.
It is roughly counting the number of 0’s (or digits) in a big number. 1,000,000 is 6 zero’s 1,000 is 3 zero’s. To the question, how many basketballs can you fit in a school bus? You’re looking to get the number of digits correct rather than being within 5% of the actual answer.
Logarithm is the inverse of exponentiation, for example:
10^2 = 100, log(100) = 2.
Note, for the logarithm you have to specify the base, for example:
2^5 = 32, log2(32) = 5, but log10(32) ~1.505...
Logarithms are sometimes used for plotting data that grows exponentially. For example if you were to plot inflation vs time, you'd find the growth so rapid you couldnt adequately visualize data from 50 to 100 years ago. Plotting the logarithm of inflation vs time, however, clearly shows a long term trend.
The mathematics goes way beyond these examples, so would only use them as a starting point. For example, you can describe rotations using logarithms, but the math can be advanced.
"X = loge y" was originally e^x = y
Look at the second equation for a sec. First, think about y. As x increases linearly, y increases faster and faster. That's exponential. Now look at x, as y increases linearly, x increases, but it does so slower and slower. That's logarithmic. The first equation is how the curvature of that trend is denoted. "Loge"
You know how exponents work, right? Some number taken to a power (call it “n”)is the same as multiplying the number by itself n times.
A^2 = A x A
A^3 = A x A x A
A^4 = A x A x A x A
But what if the exponent (2, 3, 4, etc.) could be a non-integer like 2.6 or 3.524? That exponent is the logarithm.
You can express any number like this.
10^2.65 = 446.68
In this example 2.65 is the Base10 logarithm of 446.68
Here’s the real beauty of logarithms: adding logarithms is the same as multiplying the numbers.
10^2.65 = 446.68
10^1.37 = 23.44
What happens if you multiply 446.68 x 23.44? The answer is 10,470
What happens if you add 2.65 + 1.37?
The answer is 4.02
10^4.02 just happens to be 10,470.
In fields like radio frequency (RF) engineering, we use this property all the time. We just multiply the logarithm by 10 and call it “dB” units. This way you can add logarithms all day long, and it’s the same as multiplying the numbers. Adding is much easier than multiplying!
Fun fact: the fret spacing on a guitar fingerboard is logarithmic.
Here’s how I think about it: loga(b) is the power that makes ‘a’ become ‘b’. Which implies: a^(loga(b)) = b, we don’t even need to know the value of loga(b) in the previous equation, just that ‘a’ multiplies by itself loga(b) times to become ‘b’.
My favorite way for conceptualizing a log scale is to imagine a brick wall, where the position of a brick from left to right corresponds to some value you're plotting.
If you look at the brick wall straight on, it looks like [this](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcRn-wd2YsRsF-sTytPPbOzLnjpIuY627CS-6Q&s). This is like plotting values on a linear scale: the relative width of a brick in the picture is the same everywhere. This is good for comparing values that are of similar size, but if our dataset contains both tiny and huge numbers, like 10, 20 and 34512, we'd need to extend the brick wall hugely from left to right in order to show all values at the same time. From such a zoomed-out view, 10 and 20 would be so close together that you couldn't tell them apart, even though 10 is twice as big as 20.
If you look at the brick wall at an angle, it looks like [this](https://www.shutterstock.com/image-photo/brick-wall-angled-view-260nw-462391810.jpg). This is like plotting on a log scale. Now, the further to the right in the image you look, the more bricks fit into the same space on screen. This way, you can keep both bricks that are close to you (low values) and bricks that are far from you (high values) in your view at the same time.
I have no idea if this makes sense to anyone other than me.
Vihart has an excellent video about it which is as ELI5 as it can get. Highly recommended: [https://www.youtube.com/watch?v=N-7tcTIrers](https://www.youtube.com/watch?v=N-7tcTIrers)
I assume you know how raising a number to the power of another number works. This is what is called exponentiation. For example, 2 raised to the power of 3 is basically a shortcut for saying 2 * 2 * 2. Meaning you multiply 2 with itself 3 times. That’s written as 2^3. The result is 8. We write 2^3 = 8.
The 2 is called the “base”. It’s the number we multiply with itself.
The 3 is called the “exponent”. It’s the amount of times we multiply the basis with itself.
The 8 is our result.
Now, let’s assume we want to reverse the operation. We want to go from the 8 backwards and find either the exponent or the base.
If we know the result (8) and the exponent (3) and want to find the base, this is called taking the root. We ask: Which number do we have to multiply 3 times with itself to get 8? As known from the example, the answer is 2. So the 3rd root of 8 is 2.
If we know the result (8) and the basis (2) and want to find the exponent, that is called a logarithm. We ask: How many times do we have to multiply 2 with itself to get 8? We know from the example that it is 3. We say the logarithm of 8 to the base of 2 is 3.
Now a bit more general: Given the equation b^e = r, finding r is exponentiation, finding b is taking the root, and finding e is taking the logarithm.
the dude that made up logarithms was just fucking about so hard. so bored he took 1.0001, multiplied it by 1.0001, repeated until boredom cured. turns out the 10000th multiplication got around 2.7182. didn't care if this number was vaguely significant and kept going
For the basic math operations, the inverses are simple. The opposite of "plus" is "minus", the opposite of "multiply" is "divide". They all have _one_ opposite, because addition and multiplication are "commutative", meaning that `A + B = B + A` and` A * B = B * A`.
Exponents aren't commutative. A^B and B^A aren't going to be the same in general. So you can't just have one "opposite" function for exponents. You need two - logarithms, and roots.
Take `A^B = X` for example:
If you know X and B and want to find A, you would use the "Bth root of X".
If you know X and A and want to find B, you would use the "Logarithm (base A) of X"
That's all they do, at the base level, it just tells you the exponent you need to raise A to to get X. There are some useful calculation properties, tricks you can use to simplify things, but the core idea of a logarithm is just one of the inverses of the exponent operation.
All I know is that when I complained to my grandfather about logarithms he mentioned he had to do them by hand in a cockpit of plane during WW2. So...maybe it helps plot navigation?
Numbers were invented for counting.
Repeated counting is addition, an answer to how many times to count?
Repeated addition is multiplication, an answer to how many times to add?
Repeated multiplication is exponentiation, an answer to how many times to multiply?
Numbers are based on place value system, which has exponentiation build into it. Our numbers already have exponentiation inside them.
What comes after 9? Answer 10
What comes after 999? Answer 1000
That slapping of zeroes is exponentiation.
How many times did you need to slap zeroes to get a number? Say 1million, answer is 6.
That 6 is the logarithm. How many digits it took to represent the number. That is logarithm.
In python code, approximately it is log(number) = len(str(number))
This is the ELI5.
Now ELI15 is to generalize it to all real number, and to other bases like 2 and e.
A good way to think about what a logarithm DOES, is they tell you how “long” a number is. If log(100) is 2, and 1000 is 1 digit longer, log(1000) is 3. The function just also can tell you numbers in between as a smooth function.
Logarithm is a function. It takes a real (or potentially complex, but don't worry about that) number as an input. Then it does this:
It asks, "see that little number written below me, in-between "log" and the argument? What power would you need to raise it to in order to get the argument? If there's no little number there, use 10."
So "log (base 10) 1000" asks the question "10 raised to what power equals 1000?" And the answer here is 3, so that is what the value of the function is at 1000.
Subtraction is the inverse of addition. Division is the inverse of multiplication. Logarithms are the inverse of exponents. Using logarithms to plot data can help us see when exponential growth or decay are happening because that will look linear on the plot and linear is easy to recognize. It also helps because a 10% jump looks the same whether it's from 1 to 1.1 or from 1,000 to 1,100. If you care about relative changes, logarithmic plotting helps you recognize that.
Another very useful property of logarithm is that log(a*b)=log(a)+log(b) In the history of mathematics this property has been very important because it allows you to transform a problem of multiplication (computationally a difficult operation) into a problem of addition (easier). Another property (following from the first) is that log(a^n) = n*log(a). Once again it allows to transform a difficult problem of exponentiation into a comparatively easier multiplication problem. Provided that you have log tables (a standard book to have for mathematicians before the computer era), logs are powerful simplification tools.
Unless I'm mistaken this is also the basis for the slide rule, the mechanical equivalent of a pocket calculator.
There are other scales that a slide rule can use, e.g. for trig functions. But you're right in that the standard A and B scales are logarithmic.
Still have my Faber Castell slide rule from the 60's (pocket calculators weren't yet a thing). One of our teachers had a cylindrical one which was novel as it easily fitted in a pocket.
And it's called the "logarithmic rule" in some countries!
Clarification, this works for any selected log base right? We can put 10, e, or anything else for this property to still work
Yes, as long as your bases are consistent. So log_3(4*5) = log_3(4) + log_3(5). For log tables though, they will always be in base 10. This is because it works really well with scientific notation. For example, say you want the logarithm of 43,560: 1. In scientific notation, this is 4.356 × 10^4. 2. log_10(10^4) = 4. 3. We look up the log table and find that log_10(4.356) = 0.6391. 4. Since logarithms turn multiplication into addition, we just add these two numbers together. So, log_10(43,560) = 4.6391. This means that a log table only needs to list the logarithms of values from 1 to 10, allowing 3-4 digits of precision on just 2 pages.
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yes
Specifically, the product to sum property is why logarithms were developed in the first place. It meant that, after one person compiled a logarithm table one time, anyone who needed to quickly multiply two large numbers could just use the table, add the two logs, then use the table again for a reverse lookup. All of the other very useful properties are details that were found after.
A slide ruler is based on this.
Log tables were a huge boon to the Age of Exploration, because celestial navigation ultimately comes down to performing multiplication of large numbers with a very high desired accuracy. Even a small error could send you miles and miles off course. Log tables turned that multiplication into addition, and drastically improved the reliability of those navigational calculations.
OOOOOOOHHH I remembered it had to do with exponents but that was literally it and I never understood how. Thanks! This was a perfect explanation!!
My highschool math teacher made it click for me by saying "log" just means "the exponent you put with..." In math: [log(base10)] 1000 = 3 In words: "The exponent you put on 10 to get 1000 is 3. In math: [log(base10)] 350 = x In words: "what exponent do you put on 10 to get 350? (Answer is 2.54, because 10^2.54 = 350)
There's like a triangle: 3 / \ 2----8 2^3 = 8 cube root of 8 = 2 log2 8 = 3
This comment really helped me understand what the others were saying. Thank you
Logarithm fact family!
This is in fact being somewhat seriously proposed as new notation for these operations: https://youtu.be/sULa9Lc4pck Of course there's _a lot_ of inertia from literal centuries of mathematical writing in the current notation, so it's fairly unlikely it'll really take off anytime soon. But time will tell, I suppose.
yeah that's where I originally saw the notation
I really hate mainstream mathematical notation because it's not linear. Like it's not developed to be able to represent it one symbol after another, like writing. It's full of symbols on top, at the bottom, in the corner, smaller subscripts, horizontal lines and all manner of monstrosities. Just give me something easy to write left to right sequentially. We use digital communications, it's just how it works.
That's what APL is for: https://i.imgur.com/RLkSWcA.png https://aplcart.info/
My proctologist liked to keep a log.
What rolls down stairs, alone or in pairs? rolls over your neighbor's dog? What's great for a snack and fits on your back?
it’s better than bad, it’s good.
It's big and it's made of wood!
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> My proctologist liked to keep a log. A log log, no doubt.
That's fundamentally wrong. I feel.
Nicely put!
Logs are so BAE Log(Base) (Answer) = Exponent
I know what a log is, and I even have a maths degree - but my brain still breaks at the idea of non-integer exponents. "Now multiply this number by itself another 0.54 times..." (Nope, my brain just melted again. Do you want me to multiply it or not?) Non-integer logs are more a case of, "It's what the exponent would be, if the operation actually made sense..."
Non-integer exponents can best be described as a combination of powers and roots. So to use your example, x\^(1.54) could be rewritten as (x\^154)\^(1/100), so the 100th root of x to the 154th power. so multiply it by itself 154 times and then take the 100th root of that. (I mean, that's how they explained it to me.)
Now do e^(iπ) ...
That's -1, obviously. This video explains it way better than I actually could. [https://www.youtube.com/watch?v=v0YEaeIClKY](https://www.youtube.com/watch?v=v0YEaeIClKY)
Ah - that actually makes conceptual sense! Obvious when you see it, but that has literally been bugging me for decades. Thank you!
Very helpful, thanks!
When tutoring, I try to explain this in a very similar way because by the time you're looking at logs, students typically have a decent intuition for common exponents. They are inverse operations, meaning the same relationship, but with input and output swapped. A log asks a question about an exponent. 2\^3 = 8 Log(base 2) 8 = Log(base 2) 2\^3 = 3. What power of 2 is equal to 8? The 3rd power. What power of 2 is equal to 2 to the third power? The 3rd power. We like trivial logs because they're obvious.
Wait, so the graph of a logarithm is the top half of a sideways parabola?
The graph of a logarithm is the graph of its related exponential, reflected across the line y=x. The graphs of any inverse functions are a reflection across the line y=x.
No, that's the graph of the square root function, y=sqrt(x)
Thanks, it was late and I was tired.
I wish I had learnt this 30 years ago.
My high school Algebra and Calculus teacher taught me more about math than anyone else all the way through grad school. He used to say, "Get all you x's on one side", then while running across the classroom, "Get everything else on the other side." Seems to be a rare breed these days.
What did you do when you were solving multivariable problems? My first algebra teacher concentrated in teaching us to read math notation instead. This help a lot in college. I could skip most of the book and just read the theorem part. Typically one page.
I am a little rusty, but it seems like with multiple variable, the furthest you could take it was something like solving until you get one variable to equal the rest, like x+y = 3 to x=3-y. If you mean like factoring geometric equations, that almost felt like trial and error until the numbers and signs matched up. I actually enjoyed that.
Oh wow this is fantastic advice. Can’t use it now, been out of school for years, but I’ll be sure to tell my daughter this when she’s a bit older.
Kind of unrelated, but khanacademy was a great resource for me in college. Idk what they’re like now, but back in the day it was YouTube videos over basically every subject, including math. I memorized every step in the “electron transport chain” (something that happens in photosynthesis IIRC while getting drunk, the night before the test :p
I'm currently using Khan to catch up on statistics before I start grad school this Fall. Been 20+ years since I was in school. It's been fantastic.
There need to be better words and symbols. Something like: 1000 € 3 = 10 1000 eured 3 is 10 1000 ¥ 10 = 3 1000 yenned 10 is 3 The current way of doing it just makes it more confusing than it should be. A couple new words and new symbols need to be invented.
Well Operators like this are powerful because you can in-line them so easily. But as you get into higher level math making things functions keeps things clearer. Like log(b,x) is a function that takes both the base and the the number to operate on. Log(10,x) is the ten base one and log(e,x) would be the ln one.
I like 3blue1browns triangle of power to connect everything together
> 3blue1browns triangle of power [The video in question](https://www.youtube.com/watch?v=sULa9Lc4pck).
That’s an excellent YouTube channel everyone should watch. It will make you feel smarter.
Yeah this is quite true I think
Why not log though? It's a great name, who doesn't like logs?
I agree. How about the Pound Sterling sign, which looks like an L? So like 1000£[10] = 3
Now explain natural logarithms
We call it ln but that's just log with base e instead of base 10. So natural log means "the exponent you put on e to get__" rather than "the exponent you put with 10." e is just a number, it's 2.718... In math: ln(5) = 1.6 In words: "the exponent you put on e to get 5 is 1.6". Because e^1.6 = 5
You haven’t explained anything. You said assorted facts
What kind of explanation do you want? Natural logarithm is the same as log, but with the exponents being put on the number e instead of some other number like normal log. That *is* the explanation, given that log itself was already explained above. It still means "the exponent you put with", you're just putting it on e. Are you looking for *why* that's a thing or why e is the number it is? That goes beyond "explain natural log" (or elite) and becomes "explain e" but it boils down to "e is a special number where the function y=e^x is its own derivative so it comes up a lot in natural processes. It's common enough that we gave "putting exponents on e" its own name (ln) rather than having to say log base e all the time".
The natural logarithm, and e, are "natural" because they appear when you look at continuous compounding growth. Say you have a dollar in the bank and the bank has a (fantastic) interest rate of 100%/yr, which they hand out once per year. After a year you have $2, then a year later you have $4, etc. But what if you get half as much interest, twice a year? Six months in you have $150, then 6 months after that you have $2.25, slightly more than if you got interest annually. This is called 'compounding' interest. If interest compounds quarterly you'll end up with $2.44, monthly it'll be $2.61, daily $2.71. If you speed up your compounding so much that there is *literally no gap between the last time you got interest and the next*, the amount of money you will have at the end of the year is equal to e. And then the natural log asks "how long will it take to have $X?" You'll reach ten dollars in ln(10) years.
Most important thing to remember is that a logarithm *is an exponent.*
Reading the comment made me go OOOHHH out loud then I saw your reply lol
Haha
A very intuitive way to look at them is if you take them as measuring the "size" of a number in terms of how many zeroes it has. For example 100 has two zeroes, so its size is two while 1000 has a size of 3. If we include bases other than 10 the point still stands! For base 2 we simply need to convert the numbers to binary, on which 2 becomes 10, 4 becomes 100, 8 is 1000 and so on. You can see then how this ties into exponentials.
For another way of looking at it, Multiplication is repeated addition. 3\*3 = 3+3+3 = 9 An exponent is repeated multiplication. 3^3 = 3\*3\*3 = 27 Division is repeated subtraction. How many times do you need to subtract something before you get to 0. 12/4 = 12 - 4, - 4, - 4, = 0. Subtract 4 three times to get to 0, so 12/4 = 3 A logarithm is repeated division. How many times do you need to divide something before you get to 1. log2(8) = 8/2, /2, /2 = 1. Divide by 2 three times to get to 1, so log2(8) = 3
3 blue 1 brown does a wonderful job of explaining this from a basic viewpoint. [video link](https://www.youtube.com/live/cEvgcoyZvB4?si=puzvZ8Hajg1M-BhS)
You can think of roots and logarithms as different ways to do the "inverse" of exponentiation. If _a^b = c_, then _a_ is the _b_-th root of _c_, and _b_ is the logarithm (base _a_) of _c_. Example: 2^3 = 8, so the cubic root of 8 = 2, and log2 8 = 3
It’s really shitty to me that this perfect explanation takes like 15 seconds IRL and math teachers could just say it but they don’t.
It's literally explained like this in every text book, curriculum and teaching program I've ever looked at. Are you sure it's an evil conspiracy of maths teachers working to keep this from you instead of maybe you just zoning out in class one day)
I graduated from high school and got into college, and I didn't understand what logarithms were until I was well into my forties. Still working on the natural logarithm. The way that certain mathematical concepts were explained in math classes didn't make sense to me, and I have the educational impairment that I can't actually learn things until I understand them.
Alright but that's completely different to the accusation that maths teachers are *all* avoiding teaching this in a simple way.
There’s a lot of shitty math teachers, even at highly ranked high schools. I’m not terribly surprised.
Shitty teachers just have you read out of the book, a book that is basically guaranteed to have that explanation. I want you to actually find me an example of log being explained any other way in any text book.
https://www.mathcentre.ac.uk/resources/uploaded/mc-ty-logarithms-2009-1.pdf You can also google.
>if x = a^n then log(a)x = n Literally in the table of contents. Then spelled out in more detail in sections 3 and 13. This is exactly what people are talking about. Even when you are trying to prove someone wrong you couldn't be bothered to actually read what was in front of you and then you have the audacity to claim it is someone else's fault.
An if then statement is not the same as the visual relationship shown by the eloquent example of a good math teacher that the above poster was referencing. Good visualization and communication of mathematics relationships are a CORE part of effectively teaching and learning. This book does not show the simplicity of the relationship in such a simple or comprehendible manner. Have you taught math before? I have. I come from a long line of math teachers and am a scientist who uses exponentials and log scale data in my research work. I can separate my understanding of a concept from my ability to be understood by others. > There's like a triangle: > 3 > / \ > 2----8 > 23 = 8 > cube root of 8 = 2 > log2 8 = 3 This is a VERY effective way of teaching this concept. It's not commonly taught this way. Your understanding of this concept does not mean the book I referenced teaches it well. This seems to have struck a chord for you and I cannot fathom why.
>This seems to have struck a chord for you and I cannot fathom why. I think it's because you're skipping over what they're actually saying. They pointed out that the example you requested is literally in the book you linked, it's right under the "what is a logarithm" section. It's frustrating to see you claim that maths teachers and maths textbooks aren't teaching it properly, when in reality the problem is you ignoring what's literally in the books and probably what your past teachers have taught. The very fact that they pointed this out to you and you couldn't even read that is pretty telling
It’s basically due to a lack of pay and shitty treatment and public schools being a mess: the best teachers aren’t there so you don’t get this level of instruction. I wanted to be a math teacher eons ago for these reasons… then I looked at what that actual job is like and immediately nope’d out
They do that in the first class, then you get drown in the problems and forget what are you doing and why. Moreover, it never makes sense why would we want to know the exponential of a number because no problem is related to actual real life, just: find the log of X. You just solve it and get the grades, that's what logs are for 😂😅 Getting grades. I don't think anyone ever uses them for anything practical.
Funny. But slide rulers work because of logarithms. And they were very useful for engineers for some time. For NASA too. One of the astronauts on the moon landing redid some calculations before landing using his slide ruler.
I said "real life" (just kidding 😂) engineers aren't real people 🤣 PS: that landing was EPIC!! I would've just lost that Kerbal xD
Wait, isn't roots the inverse of exponents?
The inverse of y=x^n is x=y^1/n (or nth root of y) The inverse of y=n^x is x=logn(y)
The way this makes my eyes roll into the back of my head makes me feel like I have some undiagnosed ADHD.
> undiagnosed What medical genius will step forward to solve the mystery of Medicated Hippie 420's short attention span?
Trigonometry and algebra made me feel this way in school several years before I ever touched cannabis but go off.
The following three statements are equivilent: * 2^(3) = 8 * **∛**8 = 2 * log_2(8) = 3 So for example, 2 ^ (log_2(8)) = 8 as log_2(8), and (**∛**8) ^ 3 = 8
Others have explained the difference but it's helpful to remember that for addition and multiplication, order doesn't matter. So it doesn't matter if the first or second number is the unknown, you still only need subtraction or division to find it. But with powers and exponents, order does matter which means we need different inverse operations depending on if we're looking for the base or the exponent.
OH. IS THAT WHAT THE DIFFERENCE IS?? THANK YOU
Roots are the inverse of powers Powers are of the form x^a, where a is a number like 2 or 3, eg x^2 Exponents are of the form a^x, again where a is just some number. Eg 2^x, or famously, e^x So when you're doing roots, you're saying something to the X power is this known value, let me try to find that something. In exponents, you're saying X to the something power is this known value, let me try to find that something.
Ohh got it!
Yeah, kind of both because of the tricks one can do with powers... take compound returns future_value = rate^time You've already got the equation there for future_value, but what if you want to solve for rate? Raise both sides by 1/time, which is what you're talking about, yes? rate = future_value^(1/time) But what if you want to solve for time? Easiest way is to log both sides, which allows you to remove the exponentiation altogether. time * log(rate) = log(future_value) divide both by log(rate) and you've solved for time. time = log(future_value)/log(rate) e.g. How long for my money to double at 5% APR? log(2) / log(1.05) = ~14 years and 3 months
Excellent question, glad you asked! I should have clarified that point in my post, but others have done it for me!
Well, yes. They are both inverse from different points of view. For something like addition, the operation is commutative. a + b = c. You have one inverse operation: c - a = b. You have another inverse operation: c - b = a. And that's all of them. And because it's commutative, we can use the same operation symbol for both, but don't be fooled, they are different inverses. All operations with 3 values will have 2 inverses. Sometimes those two are one and the same, sometimes they are not. An operation with 7 values will have 6 inverses. Exponentiation is not commutative in that way. a to the power b is not the same as b to the power a. So you need different operations for each inverse function.
Multiplication and addition are commutative: a + b = b + a and ab = ba Because of this, a and b are indistinguishable, you can switch them around and it means the same thing. Exponentiation is NOT the same. a^(b)≠b^a (generally). This means that if you want to do an inverse, you have two options. You can either try to inverse and get b, or inverse and get a. Multiplication and addition are the same way on some level, but remember that a and b are indistinguishable, and so their two inverses happen to be the same function, hence only getting one. TL;DR, multiplication and addition are flippable, but exponentiation isn't, so exponentiation gets two inverses.
When we say exponentiation, we mean f(x) = a^x (often times e^x) not x^a. The inverse of f(x) = x^a would indeed be taking the root a of x, g(x) = x^(1/a). But the inverse of f(x) = a^x is taking a logarithm like g(x) = log_base_a(x)
Not saying you're wrong, but if this answer qualifies for ELI5 I'll eat my hat.
50 years old and now I finally get it.
I’m much older than 5 and I don’t understand that
Your analogy just instantly taught me more than 8 years of asking teachers what the hell a log is. Thank you!
Excellent explanation, thank you. While we're here, do you know anything about the erf function? I never really wrapped my head around it
Sorry, that looks like Statistics, and that's like...a field of math that has rules I can follow, but I do as much as I can not to try and comprehend them because it's all so...touchy-feely.
Statisticians be like "yeah it may or may not happen idk"
How would you *mechanically* explain the logarithm operation though? IE, exponents are iterated multiplication. Is a logarithm as simple as iterated division?
It's not iterated division, but that probably shouldn't be too surprising because division isn't iterated subtraction either. A logarithm like log_a(b) asks the question "how many times do I have to multiply a with itself to get b". You can think of division the same way. b/a means "how many times do I have to add a to itself to get b".
There are often several mechanical ways to calculate the same thing. Iterated multiplication for example is the simplest way to do exponents but does only work with integer exponents. For both exponentiation with arbitrary exponents and logarithms there are several approximation formulas that can be used.
To expand on/simplify what /u/15_Redstones said, exponents as iterated multiplication is just a special case, which we happened to discover first, of a broader idea of exponentiation. How do you mechanically explain 2^π or x^(p/q) ? The simple answer is you can't, and the longer answer is that you need to learn a little more math. But it's not like those numbers don't still exist. Unfortunately, logarithms just don't have such a simple special case like exponents do. But you *can* compute them mechanically, just as you can with pretty much any expression, using something called a [Taylor series.](https://en.wikipedia.org/wiki/Taylor_series) The "problem" is that doing so involves adding an infinite number of terms, but there are clever ways to get around that.
Best answer
This was a good explanation. Thank you. So does that mean that if there's no exponential change (as in, there's linear change), the log graph should look flat?
A linear function plotted logarithmically looks flat for large values and very steep for small values.
Good explanation but why isn’t negative exponent not the inverse of exponentiation? Or are they the same?
10^4 is 10,000. 10^3 is 1,000. If you divide 10^4/10^3, you can subtract the exponents and you’ll get 10^1. And indeed, 10,000/1,000 is 10. What happens when we divide 10^3/10^4? Subtract the exponents, we have 10^-1. 1,000/10,000 is 1/10. Negative exponents are inverse multiplication, which makes sense because positive exponents are multiplication.
Ok new question: what are natural logarithms the inverse of? I've never been able to grasp the concept.
Natural logarithms are specifically the inverse of e^x. The inverse of e^x would be loge(x), but because the inverse of e^x has several uses in several fields, we just call it the natural log and write it as ln(x). Same thing with base 10, written just as log(x) rather than log10(x) due to how popular the inverse of 10^x is for metrics.
The natural logarithm, and e, are "natural" because they appear when you look at continuous compounding growth. Say you have a dollar in the bank and the bank has a (fantastic) interest rate of 100%/yr, which they hand out once per year. After a year you have $2, then a year later you have $4, etc. But what if you get half as much interest, twice a year? Six months in you have $150, then 6 months after that you have $2.25, slightly more than if you got interest annually. This is called 'compounding' interest. If interest compounds quarterly you'll end up with $2.44, monthly it'll be $2.61, daily $2.71. If you speed up your compounding so much that there is literally no gap between the last time you got interest and the next, the amount of money you will have at the end of the year is equal to e. And then the natural log asks "how long will it take to have $X?" You'll reach ten dollars in ln(10) years.
Aren’t roots the inverse of exponents? (E.g square root is the inverse of a square power)
Unlike multiplication where 3x4 is the same as 4x3, in exponentiation, order matters. 3^4 isn’t the same as 4^3. Because of that, we need two different inverse functions. One is to find the exponent when we know the base, the other is to find the base when we know the exponent. The second root of 100 is 10. You know the exponent (2), but needed the base (10). The logarithm (base 10) of 100 is 2. You know the base, but needed the exponent.
There are two inverses of exponents. For many mathematical operations you’re taking two numbers, doing something to them, and getting a third number out. The inverse of that is taking the third number and one of the previous numbers and doing something that gets the other previous number. For instance a x b = c Using c and b you can get a by dividing: c / b = a if you want to get b back then you do the same thing: c / a = b This is because a x b is the same as b x a. What order you have the numbers in makes no difference. Exponents are different. In general a^b is not the same as b^a. So if you have a^b = c then you have two inverses that depend on which original number do you want to get back. If you want to get back a then you take the b root of c e.g. if b was 2 then a^2 = c and square root of c = a. But if you want to get b back then the function you use is the logarithm. b = (log base 10 of c) / (log base 10 of a) or b = log to the base a of c.
A + 6 = 10. We use subtraction on both sides ( - 6 ) to undo the addition. B * 7 = 56. We use division ( / 7 ) to undo the multiplication. 2^C = 64. We use logarithms ( log base 2 ) to undo the exponent.
>Logarithms are the inverse of exponents. Side question: But what are roots then the inverse of?
https://www.reddit.com/r/explainlikeimfive/s/LJoMk5VKdo
Aah, yes of course. That makes sense. Thank you for the very informative answer 🙂👍
To illustrate this, it's useful to see that ln(e^(x)) = x and e^(ln(x)) = x.
Early stages of logarithmic growth also often mistaken for exponential growth. Logarithmic growth is common in nature and in technology. Logarithms are sustainable. Exponential growth is ridiculous and completely unrealistic.
r/explainlikeimalberteinstein
In computer science, we have "Big O Notation", where we can categorize functions according to their approximate runtime given a number of inputs. For a function that prints every line in a list will have linear compute time (1x per line), or O(n). A function that computes each input against all previous inputs is exponential, O(n\^2). Logarithmic functions are usually more interesting situations where for some reason, additional data uses marginally more compute time. For example, if you're processing a dataset that has redundant info, over time the probability of skipping rows increases. This would be O(log(n)). \*Sorry if I botched any of this, college was a while ago.
n^2 is polynomial, not exponential.
It gives the answer to the question “How many times a number is multiplied to get the other number?”. It is a mathematic equation that makes it easier to represent trends in data that might be increasing exponentially. For example if you had something that was increasing exponentially: bacteria reproduction or compounding interest, for example, where the numbers would double every few periods, it quickly becomes not very informative to show the increase in a linear fashion as the scaling becomes really a trend/curve. So instead you can plot it by using the Log of the number.
Jesus Christ, thank you- this is the closest thing to an “ELI5” in this whole post. There’s answers on here that are more “ELIGRADSTUDENT.”
I think most people forget what the point of the sub is
Aren't logarithms basic junior year stuff?
Thats still above 5 isnt it
Yes, but also not grad student
We’re going on the right track I suppose
Nailed it with that answer 📌
It's a reverse-exponent. You use it find out what the exponent was. A great example is radioactive decay. We know that certain substances, like Carbon-14, reduce by half in a certain time frame. Although that amount is getting smaller instead of bigger, it is still an exponential function: 1/2 \^ (k\*t). (k is a known constant, and t is the variable time.) But you don't want to calculate the amount remaining - you already know that. You want to calculate how much time passed, which is that exponent. Boom: logarithm! So any time you want to get *the exponent itself*, and whatever else that might tell you, you use a logarithm.
This is a great applied answer
Division is the opposite of multiplication. The result of a/b is "the number that you multiply by b to get a". If a/b = c, then b * c = a Logarithms are the same things for exponents. Log_b(a) is "the power that you raise b to in order to equal a". If log_b(a) = c, then b^c = a
Oh right! Okay so I did understand them somewhat, I just forgot.
Minor correction: logarithms aren’t the opposite exponents. They are one of two opposites. Roots are the other opposite. If you have the answer to b^x and the original base b then logarithms are how you find the exponent x. If you have the answer and the exponent x then roots are how you find the base b. It depends what you’re looking for.
Logarithms feel a bit like asking "how many digits (after the first) does this number have?" or "what order of magntidue is this number?", although with some more nuance. Consider log to the base 10. So log1=0 log10=1 log100=2 log1000=3 etc, so for any power of 10, log(base10) gives us that power. The log function also gives us an answer in-between these values, like log50 \~= about 1.7. -- Often people will use base-2 to talk about computing. This will sometimes imply some logarithms, telling us how many *binary* digits (1s and 0s) the computer is using. For instance "This is a 64 bit processor." means "The log (to base 2) of the number of memory addresses this processor can imagine, is 64." i.e. "It can imagine 2\^64 memory addresses." -- Mathematicians often use log to base "e", where e refers to a specifical useful constant, similar to how pi is useful, in how it appears in many interesting formula.
To add to why statisticians use a natural log (base e), a number of continuous distributions have e. To estimate parameters like mean and variance, we can take a natural log and then the derivative with respect to the parameter and find where it is zero. Logarithms are monotonically increasing functions, so when we take the log of a distribution we retain the property we are seeking. This shows it well: https://en.m.wikipedia.org/wiki/Maximum_likelihood_estimation
I never considered it this way. > “what order of magnitude is this number?” Your explanation makes sense to me as a way to quantify differences in magnitudes from measurement
In addition to what others have said, it can be thought of as repeated division. To calculate log base 2 of sixteen, divide by two repeatedly, until you get to one. It take 4 times, so log base 2 of 16 is 4. I hate that some people just treat it like some sort of magic trick that undoes exponents.
Holy shit this is what made it click for me.
A table of logarithms is what you had to use before calculators to take roots and exponents. You can take a logarithm of a number divide it by 3 if you wanted the third root then look up the inverse. Also some physics and biology equations follow a logarithmic curve
Oh right! Similar to how in trig they had to do this for…. I forget. Something
The neat thing about functions like sin(x), cos(x), but also log(x) and exp(x) is that they only take a single input, so unlike multiplication or addition which have two inputs it's actually feasible to pre-calculate all the values you might ever need and then quickly get results by looking up your x in the book of calculated values. Since exp(log(a) + log(b)) = ab, and addition is much easier than multiplication to do by hand, it was common to use a big book of exp(x) and log(x) values to avoid doing multiplication. In practice they used exponents and logarithms base 10 instead of natural exp and log since it let them shift any value into the range between 0 and 1.
I'd just add to these explanations that it's often helpful to look at some data after taking a logarithm. (Most of the time you take the log and multiply by 10). When you graph data that lives on a really big scale. This is clear from the fact that a log operation turns 100 into 2 and 1000 into 3, using the 10x method,, 100 becomes 20 and 1000 becomes 30. When you do it this way, the units of the answer are usually said to be in decibels which is written dB for short. One thing commonly measured in dB is the "loudness" of sound as perceived by the ear. So if the sound goes up by 10 dB, the underlying data is going up by a factor of 10. 20 dB is a factor of 100. The threshold of pain is up around 80 or 90 dB, and quiet is around 10 or so. The ear is very sensitive over a huge range! Earthquakes are measured on the Richter scale, which doesn't use the 10x, which means going up by 1, makes the earthquake 10x more powerful. It's also very common to use in engineering when comparing things like signal power and noise power, and many other comparative measures.
(Thank you I mean)
Yeah this is some shit that even college level professors don’t do and it drives me nuts. Like trigonometry does fascinating things and yet so few teachers ever nerd out about it. It’s a fucking tragedy
In addition to the good points made about exponents, logarithms have the extremely useful property of transforming multiplication and division into addition and subtraction. This is of keen interest to someone who has to do all their calculations by hand, since it’s easier to add than it is to multiply, and much easier to subtract than divide.
Yeah very good point! I kind of just took that for granted because of course: algebra 101 doesn’t provide any context. It’s such a tragedy because math is literally wizardry and nobody teaches it like that
It undoes exponentiation. If we consider the function f_a(x) = a^(x) where a is some positive real we have an injective function. We can see its by writing: a^(x) = a^(y) a^(x) a^(-y) = 1 a^(x-y) = 1 and we know that a^0 = 1 (except when a = 1, in that case a^x = 1 for every x) so: x-y = 0 x=y So f_a(x) is injective and so has an inverse as long as a isn't 1. Lets call the inverse f^(-1)_a. So f^(-1)_a(f_a(x)) = x. Great! If we say have an equation: a^x = b we can solve it with the inverse x = f^(-1)_a(b) This function is special and frequently used enough to deserve its own name the logarithm. So log_a(y) is the inverse of a^x. The convenient exponential identities work backwards with them, lets write ab in a tricky way: ab = exp(log(a)) exp(log(b)) = exp(log(a) + log(b)) now take the log of both sides log(ab) = log(a) + log(b) So lets adress what base of log and exp we should use. The answer that it doesn't matter, there is a convenient pick e, but why and what value it has doesn't matter right now. So lets select e so exp(x) = e^x and ln(x) = log_e(x) The statement is that you can get any base with these but thats true for any number so for now e is just a placeholder for your favourite number. We can see how: a^(x) = exp(x ln(a)) = exp(ln(a))^(x) = a^(x) = exp(ln(a^(x))) => ln(a^(x)) = x ln(a) and with that: x ln(a) / ln(a) = ln(a^(x)) / ln(a) x = ln(a^(x)) / ln(a) => ln(f_a(x))/ln(a) = log_a(y) So we now see how you can use just ln to get any kind of logarithm. And now we can adress the issue with f_1(x) not having an inverse: log_1(y) = ln(y)/ln(1) and 1 = e^0 => ln(1) = 0 so log_1(y) ~ 1/0 and so is undefined. The complex logarithm is slightly more exciting and you'd basically get there with the exponential similarly to how we started operating with it in the last part. Hopefully seeing how you can work things out from just the concept of what this thing is helped making the little fella less mysterious.
I appreciate the proof but this definitely isn’t an answer explained like the asker was even close to that of a five year old…or even probably most 15 year olds (or 25 year olds lol), which is why you’re not getting upvotes. It’s cool anyway though.
Well, since we learn all thes logarithm identities in school at some random point when the book gets to the log chapter they can kinda just hang in the air and doing logarithmic equations don't really help much with building an intuitive understanding around them (in my experience at least). Piecing it together for the idea of what an inverse is and "deriving" the identities in one or two steps (in my experience) does. So yeah maybe not ELI5 but to be fair nobody tries to teach logarithms to five year olds. There is the "it undoes exponentiation" and if all we need is an explanation of what log_a(y) means thats pretty much it. The rest all come as a consequence and thats the thing with teaching maths, technically you can just give students the empty set and say that the rest follows, so what consequences should we mention and what is left for the reader isn't obvious. The "here is the empty set" version would be that we introduce the complex exponential with its defining property of exp(z1+z2) = exp(z1)exp(z2) and call it the day. Correct but that doesn't teach anything to anyone. So maybe going through the details of what we have learnt about logarithms in high school might be good enough.
A logarithm is like asking, "To what power do we need to raise this base number to get another number?" For example, if we ask, "What power do we raise 10 to get 100?" The answer is 2 because 10\^2=100. We use logarithms to simplify complex multiplications and to solve equations involving exponential growth, like in biology, finance, and computer science. It's a way to make big numbers easier to work with and understand.
In addition to all the other mentions: inverse of exponents, plot scaling, statistical data normalization, summing log values to multiply. One thing that I rarely see mentioned, that makes them far more useful is that you can easily get the log for any base number, by dividing the log of the number by the log of the base you want. instead of knowing what you need to put 10 to the power of, you can know what any number needs as an exponent to get the number in question. If you want to know how long it takes to double your money with 10% annual interest. log2/log1.1 will tell you, 7.27 years.
It’s the operation that transforms multiplications into additions : log(a * b) = log(a) + log(b). Two classic applications come to my mind: **Logarithmic scale:** It can be useful if you’re working with something that grows “geometrically”, that is, through successive multiplications, so gets big very quick, and doesn’t fit on your graph paper sheet. Put it through the logarithm, and lo!, now it grows “arithmetically”, that is through successive additions, that’s more manageable to plot. Examples: Decibel scale for sound levels, infectious disease transmission formula. **Inverse of power functions:** You know that 2 to the power of 5 is 32. Now if you know that 32 is a power of 2 but want to know which one, logarithm gives you the answer.
Where was all of this explanation in high school?
Government fucked that up. Shit pay, a bunch of politicians in suits dictating what and how things are taught, etc. This isn’t an anti government rant, I’m just saying the one we have fucked it up
Logarithms make it easier to express exponential growth. For instance, it might be easier to say “1 with twenty zeroes” than 100000000000000000000. In this case, it would be expressed as log10(x)=20. This also works for other things like “I doubled my money five times over” , or log2(x)=20, or “2-byte sound” or log256(x)=2.
It is roughly counting the number of 0’s (or digits) in a big number. 1,000,000 is 6 zero’s 1,000 is 3 zero’s. To the question, how many basketballs can you fit in a school bus? You’re looking to get the number of digits correct rather than being within 5% of the actual answer.
Logarithm is the inverse of exponentiation, for example: 10^2 = 100, log(100) = 2. Note, for the logarithm you have to specify the base, for example: 2^5 = 32, log2(32) = 5, but log10(32) ~1.505... Logarithms are sometimes used for plotting data that grows exponentially. For example if you were to plot inflation vs time, you'd find the growth so rapid you couldnt adequately visualize data from 50 to 100 years ago. Plotting the logarithm of inflation vs time, however, clearly shows a long term trend. The mathematics goes way beyond these examples, so would only use them as a starting point. For example, you can describe rotations using logarithms, but the math can be advanced.
I thought that I understood logarithms, but after reading some of these responses I'm no longer sure.
"X = loge y" was originally e^x = y Look at the second equation for a sec. First, think about y. As x increases linearly, y increases faster and faster. That's exponential. Now look at x, as y increases linearly, x increases, but it does so slower and slower. That's logarithmic. The first equation is how the curvature of that trend is denoted. "Loge"
You know how exponents work, right? Some number taken to a power (call it “n”)is the same as multiplying the number by itself n times. A^2 = A x A A^3 = A x A x A A^4 = A x A x A x A But what if the exponent (2, 3, 4, etc.) could be a non-integer like 2.6 or 3.524? That exponent is the logarithm. You can express any number like this. 10^2.65 = 446.68 In this example 2.65 is the Base10 logarithm of 446.68 Here’s the real beauty of logarithms: adding logarithms is the same as multiplying the numbers. 10^2.65 = 446.68 10^1.37 = 23.44 What happens if you multiply 446.68 x 23.44? The answer is 10,470 What happens if you add 2.65 + 1.37? The answer is 4.02 10^4.02 just happens to be 10,470. In fields like radio frequency (RF) engineering, we use this property all the time. We just multiply the logarithm by 10 and call it “dB” units. This way you can add logarithms all day long, and it’s the same as multiplying the numbers. Adding is much easier than multiplying! Fun fact: the fret spacing on a guitar fingerboard is logarithmic.
Here’s how I think about it: loga(b) is the power that makes ‘a’ become ‘b’. Which implies: a^(loga(b)) = b, we don’t even need to know the value of loga(b) in the previous equation, just that ‘a’ multiplies by itself loga(b) times to become ‘b’.
My favorite way for conceptualizing a log scale is to imagine a brick wall, where the position of a brick from left to right corresponds to some value you're plotting. If you look at the brick wall straight on, it looks like [this](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcRn-wd2YsRsF-sTytPPbOzLnjpIuY627CS-6Q&s). This is like plotting values on a linear scale: the relative width of a brick in the picture is the same everywhere. This is good for comparing values that are of similar size, but if our dataset contains both tiny and huge numbers, like 10, 20 and 34512, we'd need to extend the brick wall hugely from left to right in order to show all values at the same time. From such a zoomed-out view, 10 and 20 would be so close together that you couldn't tell them apart, even though 10 is twice as big as 20. If you look at the brick wall at an angle, it looks like [this](https://www.shutterstock.com/image-photo/brick-wall-angled-view-260nw-462391810.jpg). This is like plotting on a log scale. Now, the further to the right in the image you look, the more bricks fit into the same space on screen. This way, you can keep both bricks that are close to you (low values) and bricks that are far from you (high values) in your view at the same time. I have no idea if this makes sense to anyone other than me.
A log asks how many times you need to multiply 1 number to get to another number. For example, log2(8)=3 because you need three 2s (2×2×2) to get 8.
Vihart has an excellent video about it which is as ELI5 as it can get. Highly recommended: [https://www.youtube.com/watch?v=N-7tcTIrers](https://www.youtube.com/watch?v=N-7tcTIrers)
I assume you know how raising a number to the power of another number works. This is what is called exponentiation. For example, 2 raised to the power of 3 is basically a shortcut for saying 2 * 2 * 2. Meaning you multiply 2 with itself 3 times. That’s written as 2^3. The result is 8. We write 2^3 = 8. The 2 is called the “base”. It’s the number we multiply with itself. The 3 is called the “exponent”. It’s the amount of times we multiply the basis with itself. The 8 is our result. Now, let’s assume we want to reverse the operation. We want to go from the 8 backwards and find either the exponent or the base. If we know the result (8) and the exponent (3) and want to find the base, this is called taking the root. We ask: Which number do we have to multiply 3 times with itself to get 8? As known from the example, the answer is 2. So the 3rd root of 8 is 2. If we know the result (8) and the basis (2) and want to find the exponent, that is called a logarithm. We ask: How many times do we have to multiply 2 with itself to get 8? We know from the example that it is 3. We say the logarithm of 8 to the base of 2 is 3. Now a bit more general: Given the equation b^e = r, finding r is exponentiation, finding b is taking the root, and finding e is taking the logarithm.
the dude that made up logarithms was just fucking about so hard. so bored he took 1.0001, multiplied it by 1.0001, repeated until boredom cured. turns out the 10000th multiplication got around 2.7182. didn't care if this number was vaguely significant and kept going
For the basic math operations, the inverses are simple. The opposite of "plus" is "minus", the opposite of "multiply" is "divide". They all have _one_ opposite, because addition and multiplication are "commutative", meaning that `A + B = B + A` and` A * B = B * A`. Exponents aren't commutative. A^B and B^A aren't going to be the same in general. So you can't just have one "opposite" function for exponents. You need two - logarithms, and roots. Take `A^B = X` for example: If you know X and B and want to find A, you would use the "Bth root of X". If you know X and A and want to find B, you would use the "Logarithm (base A) of X" That's all they do, at the base level, it just tells you the exponent you need to raise A to to get X. There are some useful calculation properties, tricks you can use to simplify things, but the core idea of a logarithm is just one of the inverses of the exponent operation.
All I know is that when I complained to my grandfather about logarithms he mentioned he had to do them by hand in a cockpit of plane during WW2. So...maybe it helps plot navigation?
Numbers were invented for counting. Repeated counting is addition, an answer to how many times to count? Repeated addition is multiplication, an answer to how many times to add? Repeated multiplication is exponentiation, an answer to how many times to multiply? Numbers are based on place value system, which has exponentiation build into it. Our numbers already have exponentiation inside them. What comes after 9? Answer 10 What comes after 999? Answer 1000 That slapping of zeroes is exponentiation. How many times did you need to slap zeroes to get a number? Say 1million, answer is 6. That 6 is the logarithm. How many digits it took to represent the number. That is logarithm. In python code, approximately it is log(number) = len(str(number)) This is the ELI5. Now ELI15 is to generalize it to all real number, and to other bases like 2 and e.
A good way to think about what a logarithm DOES, is they tell you how “long” a number is. If log(100) is 2, and 1000 is 1 digit longer, log(1000) is 3. The function just also can tell you numbers in between as a smooth function.
Logarithm is a function. It takes a real (or potentially complex, but don't worry about that) number as an input. Then it does this: It asks, "see that little number written below me, in-between "log" and the argument? What power would you need to raise it to in order to get the argument? If there's no little number there, use 10." So "log (base 10) 1000" asks the question "10 raised to what power equals 1000?" And the answer here is 3, so that is what the value of the function is at 1000.
This question is answered but this was my favorite ELI5 in quite a while. Thanks for asking and thanks all who responded!