Solve log_{10}(x+1)-1/2log_{10}x+log_{10}(x^2-1)

$$\log _ { 10 } ( x + 1 ) - \frac { 1 } { 2 } \log _ { 10 } x + \log _ { 10 } ( x ^ { 2 } - 1 )$$

Factor

$$\frac{\log_{10}\left(e\right)\left(2\ln(x^{2}-1)+2\ln(x+1)-\ln(x)\right)}{2}$$

Solution Steps

Factor out \frac{1}{2}.

$$\frac{2\log_{10}\left(x+1\right)-\log_{10}\left(x\right)+2\log_{10}\left(x^{2}-1\right)}{2}$$

Consider 2\ln(x+1)\times \frac{1}{\ln(10)}-\ln(x)\times \frac{1}{\ln(10)}+2\ln(x^{2}-1)\times \frac{1}{\ln(10)}. Factor out \frac{1}{\ln(10)}.

$$\frac{2\ln(x+1)-\ln(x)+2\ln(x^{2}-1)}{\ln(10)}$$

Rewrite the complete factored expression.

$$\frac{2\ln(x+1)-\ln(x)+2\ln(x^{2}-1)}{2\ln(10)}$$

Evaluate

$$\log(x^{2}-1)+\log(x+1)-\frac{\log(x)}{2}$$