Expand the Logarithmic Expression log base 7 of 3^3 square root of 8

$\log_{7} (3^{3} \sqrt{8})$

Step 1

Rewrite $\log_{7} (3^{3} \sqrt{8})$ as $\log_{7} (3^{3}) + \log_{7} (\sqrt{8})$ .

$\log_{7} (3^{3}) + \log_{7} (\sqrt{8})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8}$ as $8^{\frac{1}{2}}$ .

$\log_{7} (3^{3}) + \log_{7} (8^{\frac{1}{2}})$

Step 3

Expand $\log_{7} (3^{3})$ by moving $3$ outside the logarithm.

$3 \log_{7} (3) + \log_{7} (8^{\frac{1}{2}})$

Step 4

Expand $\log_{7} (8^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$3 \log_{7} (3) + \frac{1}{2} \log_{7} (8)$

Step 5

Rewrite $\log_{7} (8)$ as $\log_{7} (2^{3})$ .

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

Step 6

Expand $\log_{7} (2^{3})$ by moving $3$ outside the logarithm.

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

Step 7

Multiply $\frac{1}{2} (3 \log_{7} (2))$ .

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

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

Combine $\frac{3}{2}$ and $\log_{7} (2)$ .

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