Question

Write each expression as a single logarithm.

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

Short answer

The result is$-2\ln \left(x-1\right)$

Step-by-step solution

Step 1. Given information.

Consider the given information.

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

Step 2. Calculation.

Apply the logarithm property to combine the terms.

$\begin{array}{rcl}\ln \left(\frac{x}{x-1}\right)+\ln \left(\frac{x+1}{x}\right)-\ln \left(x^2-1\right)& =& \ln \left(\frac{x}{x-1}\right)+\ln \left(\frac{x+1}{x}\right)-\ln \left(x+1\right)\left(x-1\right)\\ & =& \ln x-\ln \left(x-1\right)+\ln \left(x+1\right)-\ln x-\ln \left(x+1\right)-\ln \left(x-1\right)\\ & =& -\ln \left(x-1\right)-\ln \left(x-1\right)\\ & =& -2\ln \left(x-1\right)\end{array}$

Step 3. Conclusion.

The obtained result is $-2\ln \left(x-1\right)$.