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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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    $\begingroup$ While both change of bases formulae rely on the power rule for logarithms, they apply an exponentiation and logarithm in a different order, which doesn't result in the symmetry you were looking for. At least they both include the same $\log_c a$ factor. $\endgroup$
    – jensph
    Feb 2, 2021 at 0:12

2 Answers 2

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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. $\endgroup$ Jun 21, 2019 at 15:25
  • $\begingroup$ @Peter Foreman Yes, of course! :) $\endgroup$ Jun 21, 2019 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? $\endgroup$ Jun 22, 2019 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: $(a^x)^y=a^{xy}.$ $\endgroup$ Jun 22, 2019 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. $\endgroup$ Jun 22, 2019 at 8:48
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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.

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