Precalculus Examples

$\ln (\frac{x^{8} \sqrt{p^{3} q}}{e^{2}})$

Step 1

Rewrite $\ln (\frac{x^{8} \sqrt{p^{3} q}}{e^{2}})$ as $\ln (x^{8} \sqrt{p^{3} q}) - \ln (e^{2})$ .

$\ln (x^{8} \sqrt{p^{3} q}) - \ln (e^{2})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{p^{3} q}$ as ${(p^{3} q)}^{\frac{1}{2}}$ .

$\ln (x^{8} {(p^{3} q)}^{\frac{1}{2}}) - \ln (e^{2})$

Step 3

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

$\ln (x^{8} {(p^{3} q)}^{\frac{1}{2}}) - (2 \ln (e))$

Step 4

The natural logarithm of $e$ is $1$ .

$\ln (x^{8} {(p^{3} q)}^{\frac{1}{2}}) - (2 \cdot 1)$

Step 5

Multiply $2$ by $1$ .

$\ln (x^{8} {(p^{3} q)}^{\frac{1}{2}}) - 1 \cdot 2$

Step 6

Multiply $- 1$ by $2$ .

$\ln (x^{8} {(p^{3} q)}^{\frac{1}{2}}) - 2$

Step 7

Rewrite $\ln (x^{8} {(p^{3} q)}^{\frac{1}{2}})$ as $\ln (x^{8}) + \ln ({(p^{3} q)}^{\frac{1}{2}})$ .

$\ln (x^{8}) + \ln ({(p^{3} q)}^{\frac{1}{2}}) - 2$

Step 8

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

$8 \ln (x) + \ln ({(p^{3} q)}^{\frac{1}{2}}) - 2$

Step 9

Expand $\ln ({(p^{3} q)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$8 \ln (x) + \frac{1}{2} \ln (p^{3} q) - 2$

Step 10

Rewrite $\ln (p^{3} q)$ as $\ln (p^{3}) + \ln (q)$ .

$8 \ln (x) + \frac{1}{2} (\ln (p^{3}) + \ln (q)) - 2$

Step 11

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

$8 \ln (x) + \frac{1}{2} (3 \ln (p) + \ln (q)) - 2$

Step 12

Simplify each term.

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Step 12.1

Apply the distributive property.

$8 \ln (x) + \frac{1}{2} (3 \ln (p)) + \frac{1}{2} \ln (q) - 2$

Step 12.2

Multiply $\frac{1}{2} (3 \ln (p))$ .

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Step 12.2.1

Combine $3$ and $\frac{1}{2}$ .

$8 \ln (x) + \frac{3}{2} \ln (p) + \frac{1}{2} \ln (q) - 2$

Step 12.2.2

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

$8 \ln (x) + \frac{3 \ln (p)}{2} + \frac{1}{2} \ln (q) - 2$

Step 12.3

Combine $\frac{1}{2}$ and $\ln (q)$ .

$8 \ln (x) + \frac{3 \ln (p)}{2} + \frac{\ln (q)}{2} - 2$