Answer
$4-ln(8)$
Work Step by Step
Based on the quotient rule of logarithms, we know that $log_{b}(\frac{M}{N})=log_{b}M-log_{b}N$ (where $b$, $M$, and $N$ are positive real numbers and $b\ne1$).
We know that $ln(x)$ is a natural logarithm with an understand base of $e$. Therefore, $ln(\frac{e^{4}}{8})=log_{e}e^{4}-ln(8)$.
Based on the definition of the logarithmic function, we know that $y=log_{b}x$ is equivalent to $b^{y}=x$ (for $x\gt0$ and $b\gt0$, $b\ne1$).
Therefore, $log_{e}e^{4}=4$, because $(e)^{4}=e^{4}$. So, $log_{e}e^{4}-ln(8)=4-ln(8)$.