Answer
$log_{3}(x^{4}+2x^{3})$
Work Step by Step
The power property of logarithms tells us that $log_{b}x^{r}=r log_{b}x$ (where x and b are positive real numbers, $b\ne1$, and r is a real number).
Therefore, $4log_{3}x-log_{3}x+log_{3}(x+2)=log_{3}x^{4}-log_{3}x+log_{3}(x+2)$.
The quotient property of logarithms tells us that $log_{b}\frac{x}{y}=log_{b}x-log_{b}y$ (where x, y, and, b are positive real numbers and $b\ne1$).
Therefore, $log_{3}x^{4}-log_{3}x+log_{3}(x+2)=log_{3}\frac{x^{4}}{x}+log_{3}(x+2)=log_{3}x^{3}+log_{3}(x+2)$.
The product property of logarithms tells us that $log_{b}xy=log_{b}x+log_{b}y$ (where x, y, and, b are positive real numbers and $b\ne1$).
Therefore, $log_{3}x^{3}+log_{3}(x+2)=log_{3}(x^{3}(x+2))=log_{3}(x^{4}+2x^{3})$.
