# Math Help - Logarithm Proof

1. ## Logarithm Proof

How do I prove that a^(log x with the base of a) = x? All I know is that it does equal x.

2. Originally Posted by xwrathbringerx
How do I prove that a^(log x with the base of a) = x? All I know is that it does equal x.
The logarithm function, let's call it g(x), $g(x) = \log_{a} x$, is defined as the inverse function of $f(x) = a^x$.

From the definition of an inverse function we know if must be true that
$f(f^{-1}(x))=x$

So applying that definition here

$f(g(x)) = x$
$f(\log_{a} x) = x$
$a^{\log_{a} x} = x$

And there you have it.