Is it true that x = e^ x for all real x? - Vidyadip

Question

Is it true that $x = e^{\log x}$ for all real $x?$

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Answer

First, observe that the domain of log function is a set of all positive real numbers. So the above equation is not true for non-positive real numbers.
Now, let $y = e^{\log x}.$
If $y > 0,$ we may take logarithm which gives us
$\log y = \log (e^{\log x}) = \log x . \log e = \log x.$
Thus $y = x.$
Hence $x = e^{\log x}$ is true only for positive values of $x.$

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