Logarithm proof problem: $a^{\log_b c} = c^{\log_b a}$ I have been hit with a homework problem that I just have no idea how to approach. Any help from you all is very much appreciated. Here is the problem
Prove the equation: $a^{\log_b c} = c^{\log_b a}$
Any ideas?

If you apply the logarithm with base $a$ to both sides you obtain,
$$\log_a a^{\log_b c} = \log_a c^{\log_b a}$$
$$\log_b c = \log_b a \log_a c$$
$$\frac{\log_b c}{\log_b a} = \log_a c$$
however this last equality is the change of base formula and hence is true.  Reversing the steps leads to the desired equality.
