Expand the Logarithmic Expression log base 3 of fourth root of m^5n^2

$\log_{3} (\sqrt[4]{m^{5} n^{2}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{m^{5} n^{2}}$ as ${(m^{5} n^{2})}^{\frac{1}{4}}$ .

$\log_{3} ({(m^{5} n^{2})}^{\frac{1}{4}})$

Step 2

Expand $\log_{3} ({(m^{5} n^{2})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \log_{3} (m^{5} n^{2})$

Step 3

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

$\frac{1}{4} (\log_{3} (m^{5}) + \log_{3} (n^{2}))$

Step 4

Expand $\log_{3} (m^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{4} (5 \log_{3} (m) + \log_{3} (n^{2}))$

Step 5

Expand $\log_{3} (n^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{4} (5 \log_{3} (m) + 2 \log_{3} (n))$

Step 6

Apply the distributive property.

$\frac{1}{4} (5 \log_{3} (m)) + \frac{1}{4} (2 \log_{3} (n))$

Step 7

Multiply $\frac{1}{4} (5 \log_{3} (m))$ .

Combine $5$ and $\frac{1}{4}$ .

$\frac{5}{4} \log_{3} (m) + \frac{1}{4} (2 \log_{3} (n))$

Combine $\frac{5}{4}$ and $\log_{3} (m)$ .

$\frac{5 \log_{3} (m)}{4} + \frac{1}{4} (2 \log_{3} (n))$

Step 8

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$\frac{5 \log_{3} (m)}{4} + \frac{1}{2 (2)} (2 \log_{3} (n))$

Factor $2$ out of $2 \log_{3} (n)$ .

$\frac{5 \log_{3} (m)}{4} + \frac{1}{2 (2)} (2 (\log_{3} (n)))$

Cancel the common factor.

$\frac{5 \log_{3} (m)}{4} + \frac{1}{2 \cdot 2} (2 \log_{3} (n))$

Rewrite the expression.

$\frac{5 \log_{3} (m)}{4} + \frac{1}{2} \log_{3} (n)$

Step 9

Combine $\frac{1}{2}$ and $\log_{3} (n)$ .

$\frac{5 \log_{3} (m)}{4} + \frac{\log_{3} (n)}{2}$