Expand the Logarithmic Expression log base 5 of square root of (x-1)/(x+1)

$\log_{5} (\sqrt{\frac{x - 1}{x + 1}})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x - 1}{x + 1}}$ as ${(\frac{x - 1}{x + 1})}^{\frac{1}{2}}$ .

$\log_{5} ({(\frac{x - 1}{x + 1})}^{\frac{1}{2}})$

Expand $\log_{5} ({(\frac{x - 1}{x + 1})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{5} (\frac{x - 1}{x + 1})$

Rewrite $\log_{5} (\frac{x - 1}{x + 1})$ as $\log_{5} (x - 1) - \log_{5} (x + 1)$ .

$\frac{1}{2} (\log_{5} (x - 1) - \log_{5} (x + 1))$

Apply the distributive property.

$\frac{1}{2} \log_{5} (x - 1) + \frac{1}{2} (- \log_{5} (x + 1))$

Combine $\frac{1}{2}$ and $\log_{5} (x - 1)$ .

$\frac{\log_{5} (x - 1)}{2} + \frac{1}{2} (- \log_{5} (x + 1))$

Combine $\frac{1}{2}$ and $\log_{5} (x + 1)$ .

$\frac{\log_{5} (x - 1)}{2} - \frac{\log_{5} (x + 1)}{2}$

Combine the numerators over the common denominator.

$\frac{\log_{5} (x - 1) - \log_{5} (x + 1)}{2}$