Answer
$ log_{10}\frac{(x^{3}-2x)}{(x+1)}$
Work Step by Step
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, $log_{10}x-log_{10}(x+1)+log_{10}(x^{2}-2)= log_{10}\frac{x}{(x+1)}+log_{10}(x^{2}-2)$.
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_{10}\frac{x}{(x+1)}+log_{10}(x^{2}-2)= log_{10}\frac{x}{(x+1)}\times(x^{2}-2)= log_{10}\frac{x(x^{2}-2)}{(x+1)}= log_{10}\frac{(x^{3}-2x)}{(x+1)}$.
