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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.

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

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    $\begingroup$ This is valid for $b=1$ also, but not $a,c=1$ as $\log_1{(x)}$ is undefined. $\endgroup$ – Peter Foreman Jun 21 at 15:25
  • $\begingroup$ @Peter Foreman Yes, of course! :) $\endgroup$ – Michael Rozenberg Jun 21 at 15:35
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    $\begingroup$ I'm familiar with this formula. I was wondering if there was one that doesn't explicitly involve logs (since the log formula doesn't explicitly involve exponentials). Is there something of the form $a^b = f(c^b, c^a)$, for example? $\endgroup$ – Charles Hudgins Jun 22 at 5:51
  • $\begingroup$ @Charles Hudgins You need to ask clearer, what to you want to get. You looked for the exponent form for $\log_a b = \frac{\log_c b}{\log_c a}$ and you got it: $(a^x)^y=a^{xy}.$ $\endgroup$ – Michael Rozenberg Jun 22 at 6:15
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    $\begingroup$ Is that really the corresponding rule though? It's certainly related, as it's how you prove the log property, but it doesn't take the same form. For one thing, both sides of the equation have three variables, so it doesn't really tell you how to change bases. $\endgroup$ – Charles Hudgins Jun 22 at 8:48

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