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

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $5$ by $- 1$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Multiply $2$ by $- 5$ .

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