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I've met a question whereby it asked me to show that $a^{\log_cb}=b^{\log_ca}$. I'm okay with the other logarithm questions. But I don't know how to show this question out. Can anyone give some hints or explanation for me? Thanks in advance.

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    $\begingroup$ Take $\log_c$ of both sides. $\endgroup$ – David Mitra Jun 23 '15 at 9:16
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    $\begingroup$ Please do not use titles consisting only of math expressions; these are discouraged for technical reasons -- see meta. $\endgroup$ – AlexR Jun 23 '15 at 9:24
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Note that for any base $c$ $$a^b = c^{b\log_c a}$$ You can use this here to see that both are equal to $$c^{\log_c a \log_c b}$$

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This follows from $c^{\log_c a\log_c b} = c^{\log_c b\log_c a}$.

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Take the $\log_c$ of both sides: $$ \log_c(a^{\log_cb})\qquad\log_c(b^{\log_ca}) $$ Use the property of the logarithm: $$ \log_cb\log_ca\qquad\log_ca\log_cb $$ These are equal, so also the original terms are.

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