Most exponent rules have a corresponding log rule and vice versa. For example, $a^b a^c = a^{b + c}$ and $\log_a(bc) = \log_a(b) + \log_a(c)$.
Does the change of base formula
$$
\log_a b = \frac{\log_c b}{\log_c a}
$$
have a corresponding exponent form?
Edit
I'm familiar with the fact that
$$
a^b = c^{b\log_c a}
$$
This isn't what I'm looking for, though. In the log change of base formula, there is no mention of exponentiation, only logarithms and division. I'm looking specifically for an identity that allows you to transform an expression like $a^b$ to an exponential expression with a different base, say $c$, that involves only exponentiation and elementary arithmetic operations.
Best Answer
For positives $a$, $b$ and $c$ such that $a$ and $c$ are different from $1$, we obtain: $$c^{\log_cb}=b=\left(c^{\log_ca}\right)^{\log_ab}=c^{\log_ab\log_ca}.$$ We used $$(a^x)^y=a^{xy}.$$