Question

Show that${\log }_a\left(x+\sqrt{x^2-1}\right)+{\log }_a\left(x-\sqrt{x^2-1}\right)=0$.

Short answer

The result is shown in the explanation.

Step-by-step solution

Step 1. Given information.

Consider the given information.

${\log }_a\left(x+\sqrt{x^2-1}\right)+{\log }_a\left(x-\sqrt{x^2-1}\right)=0$

Step 2. Explanation.

Consider the left side of the given equation.

${\log }_a\left(x+\sqrt{x^2-1}\right)+{\log }_a\left(x-\sqrt{x^2-1}\right)$

Step 3. Calculation.

Apply the logarithm rule to combine the left side.

$\begin{array}{rcl}{\log }_a\left(x+\sqrt{x^2-1}\right)+{\log }_a\left(x-\sqrt{x^2-1}\right)& =& {\log }_a\left(x+\sqrt{x^2-1}\right)\left(x-\sqrt{x^2-1}\right)\\ & =& {\log }_a\left(x^2-\left(x^2-1\right)\right)\\ & =& {\log }_a\left(1\right)\\ & =& 0\end{array}$

The given statement is true as the left side is equal to the right side.