Prove logarithmic identity

Question

I have to prove that $a^{\log_b(x)} = x^{\log_b(a)}$ without using the base change rule. How might I go about doing this?

Answer 1

Hints:

• take logarithms base $b$ of both sides

• $\log_b\left(c^d\right) = d\, \log_b(c)$

Answer 2

Let $\log_bx=y,x=b^y$

$x^{\log_ba}=(b^{\log_ba})^y=a^y$

The last identity can be established by setting $\log_ba=z,a=b^z=b^{\log_ba}$