Answer
$log_{5}\frac{x^{3}}{(x+1)^{2}}$
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, $2log_{5}x-2log_{5}(x+1)+log_{5}x=log_{5}x^{2}-log_{5}(x+1)^{2}+log_{5}x$.
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, $log_{5}x^{2}-log_{5}(x+1)^{2}+log_{5}x=log_{5}\frac{x^{2}}{(x+1)^{2}}+log_{5}x$.
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, $log_{5}\frac{x^{2}}{(x+1)^{2}}+log_{5}x=log_{5}\frac{(x^{2}\times x)}{(x+1)^{2}}=log_{5}\frac{x^{3}}{(x+1)^{2}}$
