Answer
$ln(\frac{x^{4}y^{7}}{z^{3}})$
Work Step by Step
According to the power rule of logarithms, we know that $log_{b}M^{p}=plog_{b}M$ (when $b$ and $M$ are positive real numbers, $b\ne1$, and $p$ is any real number).
Therefore, $4ln(x)+7ln(y)-3ln(z)=ln(x^{4})+ln(y^{7})-ln(z^{3})$.
Based on the product rule of logarithms, we know that $log_{b}(MN)=log_{b}M+log_{b}N$ (for $M\gt0$ and $N\gt0$).
Therefore, $ ln(x^{4})+ln(y^{7})-ln(z^{3})=ln(x^{4}y^{7})-ln(z^{3})$.
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 $b\ne1$).
Therefore, $ ln(x^{4}y^{7})-ln(z^{3})=ln(\frac{x^{4}y^{7}}{z^{3}})$.
In this case, the given logarithm is a natural logarithm with an understood base of $e$.