# How do you evaluate e^(lny)?

Dec 16, 2016

This expression evaluates ro $y$ for every positive real number $y$. See explanation.
$f \left(y\right) = {e}^{y}$ and $g \left(y\right) = \ln y$ are inverse functions, so both compositions: ${e}^{\ln} y$ and $\ln {e}^{y}$ evaluate to $y$ in all their domains.
Since $\ln y$ is only defined for $y > 0$ the domain of the givenexpression is also $y > 0$.
So the expression ${e}^{\ln} y$ is equal to $y$ for all positive real numbers $y$