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

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

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{{(x + 4)}^{5}}{x^{3}}}$ as ${(\frac{{(x + 4)}^{5}}{x^{3}})}^{\frac{1}{2}}$ .

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

Step 2

Expand $\log ({(\frac{{(x + 4)}^{5}}{x^{3}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 3

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

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

Step 4

Expand $\log ({(x + 4)}^{5})$ by moving $5$ outside the logarithm.

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

Step 5

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

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

Step 6

Multiply $3$ by $- 1$ .

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

Step 7

Apply the distributive property.

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

Step 8

Multiply $\frac{1}{2} (5 \log (x + 4))$ .

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

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

Combine $\frac{5}{2}$ and $\log (x + 4)$ .

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

Step 9

Multiply $\frac{1}{2} (- 3 \log (x))$ .

Combine $- 3$ and $\frac{1}{2}$ .

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

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

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

Step 10

Combine the numerators over the common denominator.

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