Property of a logarithmic function

Question

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.

Answer 1

$\log(x^{\log y}) = \log(y^{\log x}) = \log x \cdot \log y$

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

Answer 2

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

Taking log for both sides we have

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

They are equivalent!

Answer 3

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

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