Change of Base Formula for Exponents

Question

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.

Answer 1

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 $$(a^x)^y=a^{xy}.$$

Answer 2

It might help to address this conceptually. The Change of Base formula (in either context) should allow you to 'change the base' of the expression to an arbitrary base 'c'. For logarithmic functions, we can state the rule as

Divide the result by the value $\log_c(a)$.

Inverting this operation produces the rule

Multiply the input by the value $X$ (for some $X$).

So we want something that looks like $$a^b = c^{(b X)}$$ Well, it turns out that $X=\log_c(a)$ is the correct value. So this really is the `Change of Base' formula for exponential functions.