Expand the Logarithmic Expression log base 5 of cube root of x^2y

$\log_{5} (\sqrt[3]{x^{2} y})$

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

$\log_{5} ({(x^{2} y)}^{\frac{1}{3}})$

Expand $\log_{5} ({(x^{2} y)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{5} (x^{2} y)$

Rewrite $\log_{5} (x^{2} y)$ as $\log_{5} (x^{2}) + \log_{5} (y)$ .

$\frac{1}{3} (\log_{5} (x^{2}) + \log_{5} (y))$

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

$\frac{1}{3} (2 \log_{5} (x) + \log_{5} (y))$

Apply the distributive property.

$\frac{1}{3} (2 \log_{5} (x)) + \frac{1}{3} \log_{5} (y)$

Multiply $\frac{1}{3} (2 \log_{5} (x))$ .

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

$\frac{2}{3} \log_{5} (x) + \frac{1}{3} \log_{5} (y)$

Combine $\frac{2}{3}$ and $\log_{5} (x)$ .

$\frac{2 \log_{5} (x)}{3} + \frac{1}{3} \log_{5} (y)$

Combine $\frac{1}{3}$ and $\log_{5} (y)$ .

$\frac{2 \log_{5} (x)}{3} + \frac{\log_{5} (y)}{3}$