Expand the Logarithmic Expression log base 5 of square root of a^7b

$\log_{5} (\sqrt{a^{7} b})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a^{7} b}$ as ${(a^{7} b)}^{\frac{1}{2}}$ .

$\log_{5} ({(a^{7} b)}^{\frac{1}{2}})$

Expand $\log_{5} ({(a^{7} b)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{5} (a^{7} b)$

Rewrite $\log_{5} (a^{7} b)$ as $\log_{5} (a^{7}) + \log_{5} (b)$ .

$\frac{1}{2} (\log_{5} (a^{7}) + \log_{5} (b))$

Expand $\log_{5} (a^{7})$ by moving $7$ outside the logarithm.

$\frac{1}{2} (7 \log_{5} (a) + \log_{5} (b))$

Apply the distributive property.

$\frac{1}{2} (7 \log_{5} (a)) + \frac{1}{2} \log_{5} (b)$

Multiply $\frac{1}{2} (7 \log_{5} (a))$ .

Combine $7$ and $\frac{1}{2}$ .

$\frac{7}{2} \log_{5} (a) + \frac{1}{2} \log_{5} (b)$

Combine $\frac{7}{2}$ and $\log_{5} (a)$ .

$\frac{7 \log_{5} (a)}{2} + \frac{1}{2} \log_{5} (b)$

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

$\frac{7 \log_{5} (a)}{2} + \frac{\log_{5} (b)}{2}$