Precalculus Examples

$\ln (\frac{x^{9} \sqrt{x^{2} + 5}}{{(x + 5)}^{9}})$

Rewrite $\ln (\frac{x^{9} \sqrt{x^{2} + 5}}{{(x + 5)}^{9}})$ as $\ln (x^{9} \sqrt{x^{2} + 5}) - \ln ({(x + 5)}^{9})$ .

$\ln (x^{9} \sqrt{x^{2} + 5}) - \ln ({(x + 5)}^{9})$

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

$\ln (x^{9} {(x^{2} + 5)}^{\frac{1}{2}}) - \ln ({(x + 5)}^{9})$

Expand $\ln ({(x + 5)}^{9})$ by moving $9$ outside the logarithm.

$\ln (x^{9} {(x^{2} + 5)}^{\frac{1}{2}}) - (9 \ln (x + 5))$

Multiply $9$ by $- 1$ .

$\ln (x^{9} {(x^{2} + 5)}^{\frac{1}{2}}) - 9 \ln (x + 5)$

Rewrite $\ln (x^{9} {(x^{2} + 5)}^{\frac{1}{2}})$ as $\ln (x^{9}) + \ln ({(x^{2} + 5)}^{\frac{1}{2}})$ .

$\ln (x^{9}) + \ln ({(x^{2} + 5)}^{\frac{1}{2}}) - 9 \ln (x + 5)$

Expand $\ln (x^{9})$ by moving $9$ outside the logarithm.

$9 \ln (x) + \ln ({(x^{2} + 5)}^{\frac{1}{2}}) - 9 \ln (x + 5)$

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

$9 \ln (x) + \frac{1}{2} \ln (x^{2} + 5) - 9 \ln (x + 5)$

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

$9 \ln (x) + \frac{\ln (x^{2} + 5)}{2} - 9 \ln (x + 5)$

To write $9 \ln (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$9 \ln (x) \cdot \frac{2}{2} + \frac{\ln (x^{2} + 5)}{2} - 9 \ln (x + 5)$

Combine $9 \ln (x)$ and $\frac{2}{2}$ .

$\frac{9 \ln (x) \cdot 2}{2} + \frac{\ln (x^{2} + 5)}{2} - 9 \ln (x + 5)$

Combine the numerators over the common denominator.

$\frac{9 \ln (x) \cdot 2 + \ln (x^{2} + 5)}{2} - 9 \ln (x + 5)$

Multiply $2$ by $9$ .

$\frac{18 \ln (x) + \ln (x^{2} + 5)}{2} - 9 \ln (x + 5)$

To write $- 9 \ln (x + 5)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{18 \ln (x) + \ln (x^{2} + 5)}{2} - 9 \ln (x + 5) \cdot \frac{2}{2}$

Combine $- 9 \ln (x + 5)$ and $\frac{2}{2}$ .

$\frac{18 \ln (x) + \ln (x^{2} + 5)}{2} + \frac{- 9 \ln (x + 5) \cdot 2}{2}$

Combine the numerators over the common denominator.

$\frac{18 \ln (x) + \ln (x^{2} + 5) - 9 \ln (x + 5) \cdot 2}{2}$

Multiply $2$ by $- 9$ .

$\frac{18 \ln (x) + \ln (x^{2} + 5) - 18 \ln (x + 5)}{2}$