Expand the Logarithmic Expression log base 5 of 7 cube root of 11*12

$\log_{5} (7 \sqrt[3]{11 \cdot 12})$

Rewrite $\log_{5} (7 \sqrt[3]{11 \cdot 12})$ as $\log_{5} (7) + \log_{5} (\sqrt[3]{11 \cdot 12})$ .

$\log_{5} (7) + \log_{5} (\sqrt[3]{11 \cdot 12})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{11 \cdot 12}$ as ${(11 \cdot 12)}^{\frac{1}{3}}$ .

$\log_{5} (7) + \log_{5} ({(11 \cdot 12)}^{\frac{1}{3}})$

Expand $\log_{5} ({(11 \cdot 12)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\log_{5} (7) + \frac{1}{3} \log_{5} (11 \cdot 12)$

Multiply $11$ by $12$ .

$\log_{5} (7) + \frac{1}{3} \log_{5} (132)$

Rewrite $\log_{5} (132)$ as $\log_{5} (2^{2} \cdot 3 \cdot 11)$ .

$\log_{5} (7) + \frac{1}{3} \log_{5} (2^{2} \cdot 3 \cdot 11)$

Rewrite $\log_{5} (2^{2} \cdot 3 \cdot 11)$ as $\log_{5} (2^{2} \cdot 3) + \log_{5} (11)$ .

$\log_{5} (7) + \frac{1}{3} (\log_{5} (2^{2} \cdot 3) + \log_{5} (11))$

Raise $2$ to the power of $2$ .

$\log_{5} (7) + \frac{1}{3} (\log_{5} (4 \cdot 3) + \log_{5} (11))$

Multiply $4$ by $3$ .

$\log_{5} (7) + \frac{1}{3} (\log_{5} (12) + \log_{5} (11))$

Simplify each term.

Rewrite $\log_{5} (12)$ as $\log_{5} (2^{2} \cdot 3)$ .

$\log_{5} (7) + \frac{1}{3} (\log_{5} (2^{2} \cdot 3) + \log_{5} (11))$

Rewrite $\log_{5} (2^{2} \cdot 3)$ as $\log_{5} (2^{2}) + \log_{5} (3)$ .

$\log_{5} (7) + \frac{1}{3} (\log_{5} (2^{2}) + \log_{5} (3) + \log_{5} (11))$

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

$\log_{5} (7) + \frac{1}{3} (2 \log_{5} (2) + \log_{5} (3) + \log_{5} (11))$

Apply the distributive property.

$\log_{5} (7) + \frac{1}{3} (2 \log_{5} (2)) + \frac{1}{3} \log_{5} (3) + \frac{1}{3} \log_{5} (11)$

Simplify.

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

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

$\log_{5} (7) + \frac{2}{3} \log_{5} (2) + \frac{1}{3} \log_{5} (3) + \frac{1}{3} \log_{5} (11)$

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

$\log_{5} (7) + \frac{2 \log_{5} (2)}{3} + \frac{1}{3} \log_{5} (3) + \frac{1}{3} \log_{5} (11)$

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

$\log_{5} (7) + \frac{2 \log_{5} (2)}{3} + \frac{\log_{5} (3)}{3} + \frac{1}{3} \log_{5} (11)$

Combine $\frac{1}{3}$ and $\log_{5} (11)$ .

$\log_{5} (7) + \frac{2 \log_{5} (2)}{3} + \frac{\log_{5} (3)}{3} + \frac{\log_{5} (11)}{3}$