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

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Expand $\log ({(x + 6)}^{5})$ by moving $5$ outside the logarithm.

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

Step 7

Simplify each term.

Rewrite $\log (4)$ as $\log (2^{2})$ .

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

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

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

Step 8

To write $5 \log (x + 6)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 9

Combine $5 \log (x + 6)$ and $\frac{2}{2}$ .

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Multiply $2$ by $5$ .

$2 \log (2) + \frac{10 \log (x + 6) - 3 \log (x)}{2}$