Answer
$log_{10}\frac{x^{3}-2x}{x+1}$
Work Step by Step
We know 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, $log_{10}x-log_{10}(x+1)+log_{10}(x^{2}-2)=log_{10}\frac{x}{x+1}+log_{10}(x^{2}-2)$.
We know that $log_{b}xy=log_{b}x+log_{b}y$ (where $x$, $y$, and $b$ are positive real numbers and $b\ne1$).
Therefore, $log_{10}\frac{x}{x+1}+log_{10}(x^{2}-2)=log_{10}\frac{x(x^{2}-2)}{x+1}=log_{10}\frac{x^{3}-2x}{x+1}$.
