Expand the Logarithmic Expression natural log of square root of r^8s^5

$\ln (\sqrt{r^{8} s^{5}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{r^{8} s^{5}}$ as ${(r^{8} s^{5})}^{\frac{1}{2}}$ .

$\ln ({(r^{8} s^{5})}^{\frac{1}{2}})$

Step 2

Expand $\ln ({(r^{8} s^{5})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \ln (r^{8} s^{5})$

Step 3

Rewrite $\ln (r^{8} s^{5})$ as $\ln (r^{8}) + \ln (s^{5})$ .

$\frac{1}{2} (\ln (r^{8}) + \ln (s^{5}))$

Step 4

Expand $\ln (r^{8})$ by moving $8$ outside the logarithm.

$\frac{1}{2} (8 \ln (r) + \ln (s^{5}))$

Step 5

Expand $\ln (s^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{2} (8 \ln (r) + 5 \ln (s))$

Step 6

Apply the distributive property.

$\frac{1}{2} (8 \ln (r)) + \frac{1}{2} (5 \ln (s))$

Step 7

Cancel the common factor of $2$ .

Factor $2$ out of $8 \ln (r)$ .

$\frac{1}{2} (2 (4 \ln (r))) + \frac{1}{2} (5 \ln (s))$

Cancel the common factor.

$\frac{1}{2} (2 (4 \ln (r))) + \frac{1}{2} (5 \ln (s))$

Rewrite the expression.

$4 \ln (r) + \frac{1}{2} (5 \ln (s))$

Step 8

Multiply $\frac{1}{2} (5 \ln (s))$ .

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

$4 \ln (r) + \frac{5}{2} \ln (s)$

Combine $\frac{5}{2}$ and $\ln (s)$ .

$4 \ln (r) + \frac{5 \ln (s)}{2}$