Expand the Logarithmic Expression log base 8 of 8 square root of 3a^5

$\log_{8} (8 \sqrt{3 a^{5}})$

Step 1

Rewrite $\log_{8} (8 \sqrt{3 a^{5}})$ as $\log_{8} (8) + \log_{8} (\sqrt{3 a^{5}})$ .

$\log_{8} (8) + \log_{8} (\sqrt{3 a^{5}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 a^{5}}$ as ${(3 a^{5})}^{\frac{1}{2}}$ .

$\log_{8} (8) + \log_{8} ({(3 a^{5})}^{\frac{1}{2}})$

Step 3

Expand $\log_{8} ({(3 a^{5})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{8} (8) + \frac{1}{2} \log_{8} (3 a^{5})$

Step 4

Logarithm base $8$ of $8$ is $1$ .

$1 + \frac{1}{2} \log_{8} (3 a^{5})$

Step 5

Rewrite $\log_{8} (3 a^{5})$ as $\log_{8} (3) + \log_{8} (a^{5})$ .

$1 + \frac{1}{2} (\log_{8} (3) + \log_{8} (a^{5}))$

Step 6

Expand $\log_{8} (a^{5})$ by moving $5$ outside the logarithm.

$1 + \frac{1}{2} (\log_{8} (3) + 5 \log_{8} (a))$

Step 7

Simplify each term.

Apply the distributive property.

$1 + \frac{1}{2} \log_{8} (3) + \frac{1}{2} (5 \log_{8} (a))$

Combine $\frac{1}{2}$ and $\log_{8} (3)$ .

$1 + \frac{\log_{8} (3)}{2} + \frac{1}{2} (5 \log_{8} (a))$

Multiply $\frac{1}{2} (5 \log_{8} (a))$ .

Combine $5$ and $\frac{1}{2}$ .

$1 + \frac{\log_{8} (3)}{2} + \frac{5}{2} \log_{8} (a)$

Combine $\frac{5}{2}$ and $\log_{8} (a)$ .

$1 + \frac{\log_{8} (3)}{2} + \frac{5 \log_{8} (a)}{2}$