College Algebra (6th Edition)

$ln(\frac{(x+6)^{4}}{x^{3}})$
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, $4ln(x+6)-3ln(x)=ln(x+6)^{4}-ln(x^{3})$. 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, $ln(x+6)^{4}-ln(x^{3})=ln(\frac{(x+6)^{4}}{x^{3}})$. In this case, the given logarithm is a natural logarithm with an understood base of $e$.