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

$\log (8 \sqrt[4]{3 a^{5}})$

Step 1

Rewrite $\log (8 \sqrt[4]{3 a^{5}})$ as $\log (8) + \log (\sqrt[4]{3 a^{5}})$ .

$\log (8) + \log (\sqrt[4]{3 a^{5}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3 a^{5}}$ as ${(3 a^{5})}^{\frac{1}{4}}$ .

$\log (8) + \log ({(3 a^{5})}^{\frac{1}{4}})$

Step 3

Expand $\log ({(3 a^{5})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\log (8) + \frac{1}{4} \log (3 a^{5})$

Step 4

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

$\log (8) + \frac{1}{4} (\log (3) + \log (a^{5}))$

Step 5

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

$\log (8) + \frac{1}{4} (\log (3) + 5 \log (a))$

Step 6

Simplify each term.

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

$\log (2^{3}) + \frac{1}{4} (\log (3) + 5 \log (a))$

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

$3 \log (2) + \frac{1}{4} (\log (3) + 5 \log (a))$

Apply the distributive property.

$3 \log (2) + \frac{1}{4} \log (3) + \frac{1}{4} (5 \log (a))$

Combine $\frac{1}{4}$ and $\log (3)$ .

$3 \log (2) + \frac{\log (3)}{4} + \frac{1}{4} (5 \log (a))$

Multiply $\frac{1}{4} (5 \log (a))$ .

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

$3 \log (2) + \frac{\log (3)}{4} + \frac{5}{4} \log (a)$

Combine $\frac{5}{4}$ and $\log (a)$ .

$3 \log (2) + \frac{\log (3)}{4} + \frac{5 \log (a)}{4}$