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Is $x^{log(y)}$ equal to $y^{log(x)}$? If yes then how? I read it as a general property of logarithmic functions but could not understand how is it true.

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    $\begingroup$ Take logs of both. $\endgroup$ – Deepak Jul 11 '17 at 7:33
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$\log(x^{\log y}) = \log(y^{\log x}) = \log x \cdot \log y$

Now conclude, based on one-to-one nature of the log function.

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$$x^{\log y}=y^{\log x}$$

Taking log for both sides we have

$$\log y \log x = \log x \log y$$

They are equivalent!

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There is a formula: $$a=e^{\ln a}.$$

Hence: $$x^{\ln y}=e^{\ln{x^{\ln y}}}=e^{\ln y\cdot \ln x}=\left(e^{\ln y}\right)^{\ln x}=y^{\ln x}.$$

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