What is a logarithm?

Logarithms (or log for short) are highly useful in science, but they scare people a lot.


Imagine a function f(x) that takes x and does something to it.
Another function, g(x) "undoes" what f does.
In other words, 
g[f(x)] = x
In English, if you hit x with f and then hit it with g you have come right back to where you started - it's like adding two and then subtracting two from a number. In math, we call f and g inverse functions.


Now, let's define what f does. 
f(x)=10^x
So now, we define f so that it raises 10 to whatever x is.


NOW THE FUN PART! What is the g, the inverse function of f?
It is by definition a logarithm!!!!!!!!!!!!!!!!!!!
g(x)=log(x)


Let's do an example.
What is log(10^3)? Well the answer is obvious: 3! (Remember this can be rewritten as g[f(3)] which is, by definition, just 3.


Normally, logarithms are base 10 like the example, but we can define them to be base anything. Other popular logs are base e and base 2. (e is a number kinda like pi. It is roughly 2.71)