Proof that $b^{\log_b(x)} = x$

Question

I understand that the exponential functions are inverses, and would therefore map $x$ when formed as a composition, but I cannot find any formal mathmatical proofs. My thought process is: $$\log_b(b^{\log_b(x)}) = \log_b(x) \rightarrow \log_b(x)=\log_b(x) \rightarrow x=x$$ Is that the only way of going about it or are there other formal proofs?

Thanks!

Answer 1

This is a tautology since $\log_b(x)$ is defined as the exponent $m$ such that $b^m = x$.

Answer 2

By definition, if $A^x=B$, then $x=\log_AB$. Now, replace the value of x from the latter identity into the former.

Answer 3

What exponent $a$ can you put on $b$ so that $b^a=x$? $\log_b(x)$ is just the name for an $a$ that makes $b^a$ be equal to $x$. So by definition $b^{\log_b(x)}=x$.