Prove that two logarithms are equal

Question

Prove that $\log_a b^c$ is equivalent to $c * \log_a b$ with prior knowledge that:

$x = \log_a b \iff a^x = b$
$y = \log_a b^c \iff a^y = b^c$

Sorry, this was a homework question but I can't delete it since it has answers.

Answer 1

There is a typo in your second line: $y=\log_a b^c \iff a^y = b^c$. Then using these definitions of $x$ and $y$, we have $a^y=b^c=(a^x)^c=a^{cx}$, so $cx=y$.

Answer 2

we have $$a^x=b^c$$ or $$a=b^{c/x}$$ and by setting $$c\log_a b=y$$ we get $$a=b^{c/y}$$ thus we get $$b^{c/x}=b^{c/y}$$ thus we have $$y=x$$