Expand the Logarithmic Expression log of square root of a^3b

$\log (\sqrt{a^{3} b})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a^{3} b}$ as ${(a^{3} b)}^{\frac{1}{2}}$ .

$\log ({(a^{3} b)}^{\frac{1}{2}})$

Expand $\log ({(a^{3} b)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (a^{3} b)$

Rewrite $\log (a^{3} b)$ as $\log (a^{3}) + \log (b)$ .

$\frac{1}{2} (\log (a^{3}) + \log (b))$

Expand $\log (a^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{2} (3 \log (a) + \log (b))$

Apply the distributive property.

$\frac{1}{2} (3 \log (a)) + \frac{1}{2} \log (b)$

Multiply $\frac{1}{2} (3 \log (a))$ .

Combine $3$ and $\frac{1}{2}$ .

$\frac{3}{2} \log (a) + \frac{1}{2} \log (b)$

Combine $\frac{3}{2}$ and $\log (a)$ .

$\frac{3 \log (a)}{2} + \frac{1}{2} \log (b)$

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

$\frac{3 \log (a)}{2} + \frac{\log (b)}{2}$