Precalculus Examples

$\ln (\frac{18}{\sqrt{x^{6} + 1}})$

Step 1

Rewrite $\ln (\frac{18}{\sqrt{x^{6} + 1}})$ as $\ln (18) - \ln (\sqrt{x^{6} + 1})$ .

$\ln (18) - \ln (\sqrt{x^{6} + 1})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{6} + 1}$ as ${(x^{6} + 1)}^{\frac{1}{2}}$ .

$\ln (18) - \ln ({(x^{6} + 1)}^{\frac{1}{2}})$

Step 3

Expand $\ln ({(x^{6} + 1)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\ln (18) - (\frac{1}{2} \ln (x^{6} + 1))$

Step 4

Combine $\frac{1}{2}$ and $\ln (x^{6} + 1)$ .

$\ln (18) - \frac{\ln (x^{6} + 1)}{2}$

Step 5

Simplify each term.

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Step 5.1

Rewrite $\ln (18)$ as $\ln (2 \cdot 3^{2})$ .

$\ln (2 \cdot 3^{2}) - \frac{\ln (x^{6} + 1)}{2}$

Step 5.2

Rewrite $\ln (2 \cdot 3^{2})$ as $\ln (2) + \ln (3^{2})$ .

$\ln (2) + \ln (3^{2}) - \frac{\ln (x^{6} + 1)}{2}$

Step 5.3

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

$\ln (2) + 2 \ln (3) - \frac{\ln (x^{6} + 1)}{2}$

Step 6

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

$2 \ln (3) + \ln (2) \cdot \frac{2}{2} - \frac{\ln (x^{6} + 1)}{2}$

Step 7

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

$2 \ln (3) + \frac{\ln (2) \cdot 2}{2} - \frac{\ln (x^{6} + 1)}{2}$

Step 8

Combine the numerators over the common denominator.

$2 \ln (3) + \frac{\ln (2) \cdot 2 - \ln (x^{6} + 1)}{2}$

Step 9

Move $2$ to the left of $\ln (2)$ .

$2 \ln (3) + \frac{2 \ln (2) - \ln (x^{6} + 1)}{2}$