Precalculus Examples

$\ln (\sqrt{x^{8} (x + 5)})$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Apply the distributive property.

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

Step 6

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1

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

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Combine $\frac{1}{2}$ and $\ln (x + 5)$ .

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

Step 8

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

$4 \ln (x) \cdot \frac{2}{2} + \frac{\ln (x + 5)}{2}$

Step 9

Combine $4 \ln (x)$ and $\frac{2}{2}$ .

$\frac{4 \ln (x) \cdot 2}{2} + \frac{\ln (x + 5)}{2}$

Step 10

Combine the numerators over the common denominator.

$\frac{4 \ln (x) \cdot 2 + \ln (x + 5)}{2}$

Step 11

Multiply $2$ by $4$ .

$\frac{8 \ln (x) + \ln (x + 5)}{2}$