$a^b = c$, is it possible to express $b$ without logarithms?

Question

$ a^b = c $

is it possible to express b without logarithms?

Answer 1

Not without more information, since finding $b$ in this case is what logarithms are. Maybe in some special cases something can be said.

Answer 2

If $e^b=c$, then $b=\ln c$. Is it possible to express $\ln c$ without logarithms? Not unless you have a peculiar definition of "without"...

Answer 3

It is possible, certainly not the most practical though.

$5^{3x}=\dfrac{1}{125}$

$5^{3x}=\dfrac{1}{5^3}$

$5^{3x}=5^{-3}$

$3x=-3$

$\dfrac{3x}{3}=\dfrac{-3}{3}$

$\boxed{x=-1}$