An <em>inverse</em> function is any function that "undoes" another function. If we think of the function as some kind of machine that takes in a number as input and produces a number as an output, when we give inverse function the number as in input, we get , our original input, as the output
We need a function that undoes , and the natural choice for undoing an exponent is with a logarithm. Here, our base is , so we'll choose as our inverse function. Let's see how that works:
is the power we have to raise to to get , which is , so