Expand the Logarithmic Expression log of (6(x+4)^3)/( square root of x^5)

$\log (\frac{6 {(x + 4)}^{3}}{\sqrt{x^{5}}})$

Step 1

Rewrite $\log (\frac{6 {(x + 4)}^{3}}{\sqrt{x^{5}}})$ as $\log (6 {(x + 4)}^{3}) - \log (\sqrt{x^{5}})$ .

$\log (6 {(x + 4)}^{3}) - \log (\sqrt{x^{5}})$

Step 2

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

$\log (6 {(x + 4)}^{3}) - \log (x^{\frac{5}{2}})$

Step 3

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

$\log (6 {(x + 4)}^{3}) - (\frac{5}{2} \log (x))$

Step 4

Combine $\frac{5}{2}$ and $\log (x)$ .

$\log (6 {(x + 4)}^{3}) - \frac{5 \log (x)}{2}$

Step 5

Rewrite $\log (6 {(x + 4)}^{3})$ as $\log (6) + \log ({(x + 4)}^{3})$ .

$\log (6) + \log ({(x + 4)}^{3}) - \frac{5 \log (x)}{2}$

Step 6

Expand $\log ({(x + 4)}^{3})$ by moving $3$ outside the logarithm.

$\log (6) + 3 \log (x + 4) - \frac{5 \log (x)}{2}$

Step 7

Simplify each term.

Rewrite $\log (6)$ as $\log (2 (3))$ .

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

Rewrite $\log (2 (3))$ as $\log (2) + \log (3)$ .

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

Step 8

To write $3 \log (x + 4)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\log (2) + \log (3) + 3 \log (x + 4) \cdot \frac{2}{2} - \frac{5 \log (x)}{2}$

Step 9

Combine $3 \log (x + 4)$ and $\frac{2}{2}$ .

$\log (2) + \log (3) + \frac{3 \log (x + 4) \cdot 2}{2} - \frac{5 \log (x)}{2}$

Step 10

Combine the numerators over the common denominator.

$\log (2) + \log (3) + \frac{3 \log (x + 4) \cdot 2 - 5 \log (x)}{2}$

Step 11

Multiply $2$ by $3$ .

$\log (2) + \log (3) + \frac{6 \log (x + 4) - 5 \log (x)}{2}$