Precalculus Examples

$\ln (r^{8} s^{6} \sqrt[3]{r^{9} s^{2}})$

Step 1

Rewrite $\ln (r^{8} s^{6} \sqrt[3]{r^{9} s^{2}})$ as $\ln (r^{8} s^{6}) + \ln (\sqrt[3]{r^{9} s^{2}})$ .

$\ln (r^{8} s^{6}) + \ln (\sqrt[3]{r^{9} s^{2}})$

Step 2

Rewrite $\ln (r^{8} s^{6})$ as $\ln (r^{8}) + \ln (s^{6})$ .

$\ln (r^{8}) + \ln (s^{6}) + \ln (\sqrt[3]{r^{9} s^{2}})$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{r^{9} s^{2}}$ as ${(r^{9} s^{2})}^{\frac{1}{3}}$ .

$\ln (r^{8}) + \ln (s^{6}) + \ln ({(r^{9} s^{2})}^{\frac{1}{3}})$

Step 4

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

$8 \ln (r) + \ln (s^{6}) + \ln ({(r^{9} s^{2})}^{\frac{1}{3}})$

Step 5

Expand $\ln (s^{6})$ by moving $6$ outside the logarithm.

$8 \ln (r) + 6 \ln (s) + \ln ({(r^{9} s^{2})}^{\frac{1}{3}})$

Step 6

Expand $\ln ({(r^{9} s^{2})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$8 \ln (r) + 6 \ln (s) + \frac{1}{3} \ln (r^{9} s^{2})$

Step 7

Rewrite $\ln (r^{9} s^{2})$ as $\ln (r^{9}) + \ln (s^{2})$ .

$8 \ln (r) + 6 \ln (s) + \frac{1}{3} (\ln (r^{9}) + \ln (s^{2}))$

Step 8

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

$8 \ln (r) + 6 \ln (s) + \frac{1}{3} (9 \ln (r) + \ln (s^{2}))$

Step 9

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

$8 \ln (r) + 6 \ln (s) + \frac{1}{3} (9 \ln (r) + 2 \ln (s))$

Step 10

Simplify each term.

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Step 10.1

Apply the distributive property.

$8 \ln (r) + 6 \ln (s) + \frac{1}{3} (9 \ln (r)) + \frac{1}{3} (2 \ln (s))$

Step 10.2

Cancel the common factor of $3$ .

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Step 10.2.1

Factor $3$ out of $9 \ln (r)$ .

$8 \ln (r) + 6 \ln (s) + \frac{1}{3} (3 (3 \ln (r))) + \frac{1}{3} (2 \ln (s))$

Step 10.2.2

Cancel the common factor.

$8 \ln (r) + 6 \ln (s) + \frac{1}{3} (3 (3 \ln (r))) + \frac{1}{3} (2 \ln (s))$

Step 10.2.3

Rewrite the expression.

$8 \ln (r) + 6 \ln (s) + 3 \ln (r) + \frac{1}{3} (2 \ln (s))$

Step 10.3

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

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Step 10.3.1

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

$8 \ln (r) + 6 \ln (s) + 3 \ln (r) + \frac{2}{3} \ln (s)$

Step 10.3.2

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

$8 \ln (r) + 6 \ln (s) + 3 \ln (r) + \frac{2 \ln (s)}{3}$

Step 11

Add $8 \ln (r)$ and $3 \ln (r)$ .

$11 \ln (r) + 6 \ln (s) + \frac{2 \ln (s)}{3}$

Step 12

To write $6 \ln (s)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$11 \ln (r) + 6 \ln (s) \cdot \frac{3}{3} + \frac{2 \ln (s)}{3}$

Step 13

Combine $6 \ln (s)$ and $\frac{3}{3}$ .

$11 \ln (r) + \frac{6 \ln (s) \cdot 3}{3} + \frac{2 \ln (s)}{3}$

Step 14

Combine the numerators over the common denominator.

$11 \ln (r) + \frac{6 \ln (s) \cdot 3 + 2 \ln (s)}{3}$

Step 15

Simplify the numerator.

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Step 15.1

Multiply $3$ by $6$ .

$11 \ln (r) + \frac{18 \ln (s) + 2 \ln (s)}{3}$

Step 15.2

Add $18 \ln (s)$ and $2 \ln (s)$ .

$11 \ln (r) + \frac{20 \ln (s)}{3}$