Calculus Examples

$\ln (\frac{9 x}{\sqrt{x^{2} - 4}})$

Step 1

Rewrite $\ln (\frac{9 x}{\sqrt{x^{2} - 4}})$ as $\ln (9 x) - \ln (\sqrt{x^{2} - 4})$ .

$\ln (9 x) - \ln (\sqrt{x^{2} - 4})$

Step 2

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

$\ln (9 x) - \ln ({(x^{2} - 4)}^{\frac{1}{2}})$

Step 3

Expand $\ln ({(x^{2} - 4)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\ln (9 x) - (\frac{1}{2} \ln (x^{2} - 4))$

Step 4

Combine $\frac{1}{2}$ and $\ln (x^{2} - 4)$ .

$\ln (9 x) - \frac{\ln (x^{2} - 4)}{2}$

Step 5

Rewrite $\ln (9 x)$ as $\ln (9) + \ln (x)$ .

$\ln (9) + \ln (x) - \frac{\ln (x^{2} - 4)}{2}$

Step 6

Simplify each term.

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Step 6.1

Rewrite $\ln (9)$ as $\ln (3^{2})$ .

$\ln (3^{2}) + \ln (x) - \frac{\ln (x^{2} - 4)}{2}$

Step 6.2

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

$2 \ln (3) + \ln (x) - \frac{\ln (x^{2} - 4)}{2}$

Step 7

To write $2 \ln (3)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 \ln (3) \cdot \frac{2}{2} - \frac{\ln (x^{2} - 4)}{2} + \ln (x)$

Step 8

Combine $2 \ln (3)$ and $\frac{2}{2}$ .

$\frac{2 \ln (3) \cdot 2}{2} - \frac{\ln (x^{2} - 4)}{2} + \ln (x)$

Step 9

Combine the numerators over the common denominator.

$\frac{2 \ln (3) \cdot 2 - \ln (x^{2} - 4)}{2} + \ln (x)$

Step 10

Multiply $2$ by $2$ .

$\frac{4 \ln (3) - \ln (x^{2} - 4)}{2} + \ln (x)$