Expand the Logarithmic Expression log base b of square root of xy

$\log_{b} (\sqrt{x y})$

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

$\log_{b} ({(x y)}^{\frac{1}{2}})$

Expand $\log_{b} ({(x y)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{b} (x y)$

Rewrite $\log_{b} (x y)$ as $\log_{b} (x) + \log_{b} (y)$ .

$\frac{1}{2} (\log_{b} (x) + \log_{b} (y))$

Apply the distributive property.

$\frac{1}{2} \log_{b} (x) + \frac{1}{2} \log_{b} (y)$

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

$\frac{\log_{b} (x)}{2} + \frac{1}{2} \log_{b} (y)$

Combine $\frac{1}{2}$ and $\log_{b} (y)$ .

$\frac{\log_{b} (x)}{2} + \frac{\log_{b} (y)}{2}$