Log variable base with variable

Question

Looking for help with this equation. Trying to help boyfriends younger sister but answer is either all numbers or its impossible: $$ \log_x \left(x^5\right) = 5 $$

Answer 1

$\log_b a$ is really asking to which power do we need to raise $b$ to get $a$, in other words $\log_b a = x \iff b^x = a$. Therefore, $$ \log_b \left(b^5\right) = 5 $$ for all $b$ where the log is defined.

Answer 2

Remember the fundamental rule of $\log$s. $$\log_a(b)=c\iff a^c=b$$

It also requires $b>0$.

In other words, you seek all $b>0$ such that $b^5=b^5$. This is trivially all $b>0$