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Similar to how $log(xy) = log(x) + log(y)$, does a nontrivial function exist that has the property $f(x^y) = f(x)f(y)$? How would one attempt to derive such a function?

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    $\begingroup$ I don't understand the analogy here? $\endgroup$ – copper.hat Nov 15 '13 at 4:19
  • $\begingroup$ @copper.hat I guess the idea is to replace multiplication with exponentiation and addition with multiplication. $\endgroup$ – Karl Kroningfeld Nov 15 '13 at 4:21
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    $\begingroup$ Exponentiation is not analogous to multiplication. The latter is symmetrical algebraic operation, the former is not. $\endgroup$ – anon Nov 15 '13 at 4:22
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Then $f(x^y) = f(y^x) \Rightarrow f(1^y) = f(y^1) \Rightarrow f(1) = f(y)$ for any $y$. f is a constant function, $0$ or $1$.

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  • $\begingroup$ You missed an $f$. $\endgroup$ – Bennett Gardiner Nov 15 '13 at 4:48
  • $\begingroup$ Just a little typo: cons${\Large\bf\mbox{t}}$ant. It's a fine proof. UpVote $0$ k. $\endgroup$ – Felix Marin Nov 15 '13 at 5:10
  • $\begingroup$ Oh, two typos in one sentence :) $\endgroup$ – gukoff Nov 15 '13 at 5:19

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