Precalculus Examples

$\ln (\frac{x^{7} \sqrt{p^{5} q^{3}}}{e^{9}})$

Step 1

Rewrite $\ln (\frac{x^{7} \sqrt{p^{5} q^{3}}}{e^{9}})$ as $\ln (x^{7} \sqrt{p^{5} q^{3}}) - \ln (e^{9})$ .

$\ln (x^{7} \sqrt{p^{5} q^{3}}) - \ln (e^{9})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{p^{5} q^{3}}$ as ${(p^{5} q^{3})}^{\frac{1}{2}}$ .

$\ln (x^{7} {(p^{5} q^{3})}^{\frac{1}{2}}) - \ln (e^{9})$

Step 3

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

$\ln (x^{7} {(p^{5} q^{3})}^{\frac{1}{2}}) - (9 \ln (e))$

Step 4

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

$\ln (x^{7} {(p^{5} q^{3})}^{\frac{1}{2}}) - (9 \cdot 1)$

Step 5

Multiply $9$ by $1$ .

$\ln (x^{7} {(p^{5} q^{3})}^{\frac{1}{2}}) - 1 \cdot 9$

Step 6

Multiply $- 1$ by $9$ .

$\ln (x^{7} {(p^{5} q^{3})}^{\frac{1}{2}}) - 9$

Step 7

Rewrite $\ln (x^{7} {(p^{5} q^{3})}^{\frac{1}{2}})$ as $\ln (x^{7}) + \ln ({(p^{5} q^{3})}^{\frac{1}{2}})$ .

$\ln (x^{7}) + \ln ({(p^{5} q^{3})}^{\frac{1}{2}}) - 9$

Step 8

Expand $\ln (x^{7})$ by moving $7$ outside the logarithm.

$7 \ln (x) + \ln ({(p^{5} q^{3})}^{\frac{1}{2}}) - 9$

Step 9

Expand $\ln ({(p^{5} q^{3})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$7 \ln (x) + \frac{1}{2} \ln (p^{5} q^{3}) - 9$

Step 10

Rewrite $\ln (p^{5} q^{3})$ as $\ln (p^{5}) + \ln (q^{3})$ .

$7 \ln (x) + \frac{1}{2} (\ln (p^{5}) + \ln (q^{3})) - 9$

Step 11

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

$7 \ln (x) + \frac{1}{2} (5 \ln (p) + \ln (q^{3})) - 9$

Step 12

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

$7 \ln (x) + \frac{1}{2} (5 \ln (p) + 3 \ln (q)) - 9$

Step 13

Simplify each term.

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Step 13.1

Apply the distributive property.

$7 \ln (x) + \frac{1}{2} (5 \ln (p)) + \frac{1}{2} (3 \ln (q)) - 9$

Step 13.2

Multiply $\frac{1}{2} (5 \ln (p))$ .

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Step 13.2.1

Combine $5$ and $\frac{1}{2}$ .

$7 \ln (x) + \frac{5}{2} \ln (p) + \frac{1}{2} (3 \ln (q)) - 9$

Step 13.2.2

Combine $\frac{5}{2}$ and $\ln (p)$ .

$7 \ln (x) + \frac{5 \ln (p)}{2} + \frac{1}{2} (3 \ln (q)) - 9$

Step 13.3

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

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Step 13.3.1

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

$7 \ln (x) + \frac{5 \ln (p)}{2} + \frac{3}{2} \ln (q) - 9$

Step 13.3.2

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

$7 \ln (x) + \frac{5 \ln (p)}{2} + \frac{3 \ln (q)}{2} - 9$