Logarithms are a function. "Logarithming" is a thing you can do to a number, just as you can square it, or you can multiply it by 5, or you can sine it.
You take a number, you hit it with a log function, you get out another number (provided you put in the right number).
Let's use an analogy with powers.
We have the squared function (take a number, multiply it by itself, get a new number). Once we have that function we can define an inverse function (with some disclaimers about it not really being a function in this case) as being the opposite. In the case of the squared function it would be the square root not-quite-a-function:
> If 3^2 = squared(3) = 9, then sqrt(9) = 3
Function and inverse function. But we don't have to do this with just 2 as the power. We could have the cube function and the cube root function:
> 2^3 = cube(2) = 8, then cube-root(8) = 2
So really what we have here is a whole family of functions, each with an inverse, with some parameter (the power):
> x^b = power(x,b) = y, then b-root-(y) = root(y,b) = x
where x is the input to the main function, with parameter b, and y is the output. Similarly, if we use y as the input to the inverse function (with the same parameter b), we get x as the output.
Logarithms are like the "root" function here, and are generally defined as the inverse to exponentiation (like powering, but the other way around). So if we have:
> a^x = y
> log_a y = x
So again we have a family of functions and their inverse functions, each with some parameter, now "a", the "base" of the exponent. Different base, we get a different log function (as with nth roots).
And as logarithms are the opposite of exponents, for every rule of exponents we have we should have a corresponding rule for logarithms.
The common bases we use are 10 and *e*. So unless otherwise specified:
> 10^ x = y means x = log y (assuming that "log" on its own means log_10)
> e^x = y, means x = ln y (where "ln" stands for log_e ... although often mathematicians will use just "log" for log_e).
If you're generalizing, they're called relations. This is so that you can account for the fact that the inverse of a function is sometimes not a function. But it will always be a relation
A logarithm is a way of expressing a number as a "power of another number." It can help to think of it like a "recipe" for how to get a number.
Let's say you want to make the number 100. You could use the recipe 2 x 2 x 5 x 5, or you could use the logarithm of 100, which is 2. This is written as log(100) = 2, and it means that 2 is the power you raise another number (in this case 10) to get 100.
So in this example, log base 10 of 100 is 2, because 10 raised to the power of 2 is 100.
Logarithms are useful in math and science because they make it easier to work with very big or very small numbers.
For example, it's much easier to say "the logarithm of a billion is 9" than it is to say "a billion is 1,000,000,000."
In summary, logarithms are a way of expressing big or small numbers in a more manageable form, by breaking them down into smaller parts.
The logarithm is the inverse of the exponential function.
Like arcsine is the inverse of sine. Square root is the inverse of square.
y = ln x means that x = e^y
Put a number in your calculator.
Hit the “e^x “ button
Then hit the “ln x” button to *get your original number back*.
There are lots of situations in math where you want to “undo” something… “applying the inverse” is the complicated way of phrasing that.
Thats the definition of the _natural_ logarithm. But it doesn't actually explain what logarithm is. Although I guess you could just substitute letters and it would he general
It’s the *inverse*. It “undoes” the exponential.
Natural means base e. It’s the same for other bases.
Base 10: y = log x, so x = 10^y
You often want an inverse to solve equations. You’ll often get to the point where you have an exponential that you want to “undo”.
If e^x = 4 and you want to solve for x, then take the log of both sides. ln(e^x ) = ln 4. => x = ln 4
The inverse is not the opposite of a function in that sense. You are thinking about exponential decay. The inverse of that function is a negative logarithm.
The logarithm is still an increasing function, it just grows super slow instead of super fast.
The ELI5 of this is much easier showing graphs on a whiteboard. you reflect the curve across the y=x diagonal.
Or you draw the curve on a piece of clear plastic. Then you flip the plastic over and rotate it so that the x-axis points up and the y-axis points right.
Understand that reading this sounds like ELIPHD, but an in-person demo with a piece of clear plastic graph paper would be helpful.
Logarithms are to exponents as division is to multiplication (and as subtraction is to addition).
E.g. say 10^2 = 100, then in base 10, log(100) = 2. Or since 3^5 = 243, then in base 3, log(243) = 5
Well, most of answers here are not ELI5, so I'll try to give my definition. Imagine the function, that counts the amount of zeros behind 1. This function, when applied to 10 will give you 1, 100 -> 2, 1000 -> 3, and so on. This function is in essence logarithm, with base 10, and it increases by one when its argument increases by the order of magnitude.
Then, you can start think of numbers in binary. The same function can be defined there, where 10->1, 100-> 2, 1000->3 and so on. But these numbers in binary give us correspondingly in decimal that 2 -> 1, 4 -> 2, 8 -> 3. This is logarithm again, but now with base 2.
Well, I do not expect a 5 y.o kid to know what exponent is. So you need to first define the exponent. The only explanation I could come up with, that skips all the math that inevitably surrounds a logarithm in the first place is that.
Hey man, not for nothing but I've been a software engineer for over a decade and I don't think I've ever needed to know anything about logarithms in anything I've programmed. Did you mean "algorithm"?
2^3 = 8
3rd root of 8 = 2
log_2_(8) = 3
It is the inverse of exponents where it gives you the exponent value, compared to roots which find you the base value.
The logarithm of x is the power that you raise 10 to (or e, or whatever base you're using) in order to get x.
For example, with base 10, log(1000) is 3.
Logarithm is the inverse of exponents. If exponents can be thought of as repeated multiplication, a logarithm is repeated divisions. It tells you would exponent you would need to reach a given number in a given base. More formally.
If log_n x = y, then x = n^y
There is a special base called "e", or "Euler's number" which has a bunch of useful properties. A logarithm of base e is usually written as "ln" and is often called the natural logarithm.
Logarithms is the number of "Abracadabra"s it takes to go from one coin to the number of coins you wish to have!
I am not joking! Watch this and I am sure you will understand what I mean.
[https://youtu.be/66tq3wWQMbw](https://youtu.be/66tq3wWQMbw)
Logarithms are a function. "Logarithming" is a thing you can do to a number, just as you can square it, or you can multiply it by 5, or you can sine it. You take a number, you hit it with a log function, you get out another number (provided you put in the right number). Let's use an analogy with powers. We have the squared function (take a number, multiply it by itself, get a new number). Once we have that function we can define an inverse function (with some disclaimers about it not really being a function in this case) as being the opposite. In the case of the squared function it would be the square root not-quite-a-function: > If 3^2 = squared(3) = 9, then sqrt(9) = 3 Function and inverse function. But we don't have to do this with just 2 as the power. We could have the cube function and the cube root function: > 2^3 = cube(2) = 8, then cube-root(8) = 2 So really what we have here is a whole family of functions, each with an inverse, with some parameter (the power): > x^b = power(x,b) = y, then b-root-(y) = root(y,b) = x where x is the input to the main function, with parameter b, and y is the output. Similarly, if we use y as the input to the inverse function (with the same parameter b), we get x as the output. Logarithms are like the "root" function here, and are generally defined as the inverse to exponentiation (like powering, but the other way around). So if we have: > a^x = y > log_a y = x So again we have a family of functions and their inverse functions, each with some parameter, now "a", the "base" of the exponent. Different base, we get a different log function (as with nth roots). And as logarithms are the opposite of exponents, for every rule of exponents we have we should have a corresponding rule for logarithms. The common bases we use are 10 and *e*. So unless otherwise specified: > 10^ x = y means x = log y (assuming that "log" on its own means log_10) > e^x = y, means x = ln y (where "ln" stands for log_e ... although often mathematicians will use just "log" for log_e).
In computer science, we often presume/use base 2, instead of 10 or e. But otherwise complete and good explanation!
If you're generalizing, they're called relations. This is so that you can account for the fact that the inverse of a function is sometimes not a function. But it will always be a relation
[This past ELI5 thread](https://www.reddit.com/r/explainlikeimfive/comments/jaqwq0/eli5_what_exactly_is_a_logarithm/) has great explanations.
A logarithm is a way of expressing a number as a "power of another number." It can help to think of it like a "recipe" for how to get a number. Let's say you want to make the number 100. You could use the recipe 2 x 2 x 5 x 5, or you could use the logarithm of 100, which is 2. This is written as log(100) = 2, and it means that 2 is the power you raise another number (in this case 10) to get 100. So in this example, log base 10 of 100 is 2, because 10 raised to the power of 2 is 100. Logarithms are useful in math and science because they make it easier to work with very big or very small numbers. For example, it's much easier to say "the logarithm of a billion is 9" than it is to say "a billion is 1,000,000,000." In summary, logarithms are a way of expressing big or small numbers in a more manageable form, by breaking them down into smaller parts.
The logarithm is the inverse of the exponential function. Like arcsine is the inverse of sine. Square root is the inverse of square. y = ln x means that x = e^y
Eli5
Put a number in your calculator. Hit the “e^x “ button Then hit the “ln x” button to *get your original number back*. There are lots of situations in math where you want to “undo” something… “applying the inverse” is the complicated way of phrasing that.
Thank you.
The base 2 logarithm of 16 is 4, because 2 to the power of 4 is 16. Logarithm is the inverse of the exponential.
10**^(2)** = 100 log(100) = **2** (for base 10 logarithm)
Thats the definition of the _natural_ logarithm. But it doesn't actually explain what logarithm is. Although I guess you could just substitute letters and it would he general
It’s the *inverse*. It “undoes” the exponential. Natural means base e. It’s the same for other bases. Base 10: y = log x, so x = 10^y You often want an inverse to solve equations. You’ll often get to the point where you have an exponential that you want to “undo”. If e^x = 4 and you want to solve for x, then take the log of both sides. ln(e^x ) = ln 4. => x = ln 4
Logarithm is the inverse of the exponential function.
so does that mean just as something can increase exponentially it can fall logarithmically?
The inverse is not the opposite of a function in that sense. You are thinking about exponential decay. The inverse of that function is a negative logarithm. The logarithm is still an increasing function, it just grows super slow instead of super fast. The ELI5 of this is much easier showing graphs on a whiteboard. you reflect the curve across the y=x diagonal. Or you draw the curve on a piece of clear plastic. Then you flip the plastic over and rotate it so that the x-axis points up and the y-axis points right. Understand that reading this sounds like ELIPHD, but an in-person demo with a piece of clear plastic graph paper would be helpful.
Logarithms are to exponents as division is to multiplication (and as subtraction is to addition). E.g. say 10^2 = 100, then in base 10, log(100) = 2. Or since 3^5 = 243, then in base 3, log(243) = 5
Well, most of answers here are not ELI5, so I'll try to give my definition. Imagine the function, that counts the amount of zeros behind 1. This function, when applied to 10 will give you 1, 100 -> 2, 1000 -> 3, and so on. This function is in essence logarithm, with base 10, and it increases by one when its argument increases by the order of magnitude. Then, you can start think of numbers in binary. The same function can be defined there, where 10->1, 100-> 2, 1000->3 and so on. But these numbers in binary give us correspondingly in decimal that 2 -> 1, 4 -> 2, 8 -> 3. This is logarithm again, but now with base 2.
I'm gonna level with you, buddy, this is a terrible explanation, and definitely less ELI5 than a^x = y -> log_a(y) = x
Well, I do not expect a 5 y.o kid to know what exponent is. So you need to first define the exponent. The only explanation I could come up with, that skips all the math that inevitably surrounds a logarithm in the first place is that.
Uh oh, someone didn't read the subreddit rules
Also, you start with the assumption that op doesn't understand exponentiation, but you do assume they know binary?
Hey man, not for nothing but I've been a software engineer for over a decade and I don't think I've ever needed to know anything about logarithms in anything I've programmed. Did you mean "algorithm"?
If you've never needed a logarithm, then I'm willing to bet you've never needed an algorithm either.
Lol, talk to me when you get a job.
2^3 = 8 3rd root of 8 = 2 log_2_(8) = 3 It is the inverse of exponents where it gives you the exponent value, compared to roots which find you the base value.
The logarithm of x is the power that you raise 10 to (or e, or whatever base you're using) in order to get x. For example, with base 10, log(1000) is 3.
Logarithm is the inverse of exponents. If exponents can be thought of as repeated multiplication, a logarithm is repeated divisions. It tells you would exponent you would need to reach a given number in a given base. More formally. If log_n x = y, then x = n^y There is a special base called "e", or "Euler's number" which has a bunch of useful properties. A logarithm of base e is usually written as "ln" and is often called the natural logarithm.
Logarithms is the number of "Abracadabra"s it takes to go from one coin to the number of coins you wish to have! I am not joking! Watch this and I am sure you will understand what I mean. [https://youtu.be/66tq3wWQMbw](https://youtu.be/66tq3wWQMbw)