How do I prove that $a^{\log_a(b)}=b$?

Question

All in the title basically. Not sure how to prove $a^{\log_a(b)}=b$.

Don't know how to use the rules to get that.

Answer 1

The definition of logarithm will do

You may also use the following argument

Let $a^x=b$ and solve for $x$

Take logarithm and you get $\log a^x=\log b$

$$x \log a =\log b$$

$$x=\frac {\log b}{\log a} =\log _a b$$