Question

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

Answer 1

$y=e^{\log x}$

For $x=0$,

$y=e^{\log 0}$

But $\log 0$ is not defined.

Hence, the equation is not defined for $x=0$.

For $x<0$

$y=e^{\log x}$

But $\log x$ is not defined for negative numbers.

Hence, equation is not defined for $x<0$.

For $x>0$

$y=e^{\log x}$

This is defined.

Hence, equation is defined for $x>0$.

Therefore, $y=e^{\log x}$ is true for all positive values of x, i.e., $x>0$.