Precalculus Examples

$\ln (\sqrt[3]{6 r^{5} s})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6 r^{5} s}$ as ${(6 r^{5} s)}^{\frac{1}{3}}$ .

$\ln ({(6 r^{5} s)}^{\frac{1}{3}})$

Step 2

Expand $\ln ({(6 r^{5} s)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \ln (6 r^{5} s)$

Step 3

Rewrite $\ln (6 r^{5} s)$ as $\ln (6 r^{5}) + \ln (s)$ .

$\frac{1}{3} (\ln (6 r^{5}) + \ln (s))$

Step 4

Rewrite $\ln (6 r^{5})$ as $\ln (6) + \ln (r^{5})$ .

$\frac{1}{3} (\ln (6) + \ln (r^{5}) + \ln (s))$

Step 5

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

$\frac{1}{3} (\ln (6) + 5 \ln (r) + \ln (s))$

Step 6

Simplify each term.

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Step 6.1

Rewrite $\ln (6)$ as $\ln (2 (3))$ .

$\frac{1}{3} (\ln (2 (3)) + 5 \ln (r) + \ln (s))$

Step 6.2

Rewrite $\ln (2 (3))$ as $\ln (2) + \ln (3)$ .

$\frac{1}{3} (\ln (2) + \ln (3) + 5 \ln (r) + \ln (s))$

Step 7

Apply the distributive property.

$\frac{1}{3} \ln (2) + \frac{1}{3} \ln (3) + \frac{1}{3} (5 \ln (r)) + \frac{1}{3} \ln (s)$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{3}$ and $\ln (2)$ .

$\frac{\ln (2)}{3} + \frac{1}{3} \ln (3) + \frac{1}{3} (5 \ln (r)) + \frac{1}{3} \ln (s)$

Step 8.2

Combine $\frac{1}{3}$ and $\ln (3)$ .

$\frac{\ln (2)}{3} + \frac{\ln (3)}{3} + \frac{1}{3} (5 \ln (r)) + \frac{1}{3} \ln (s)$

Step 8.3

Multiply $\frac{1}{3} (5 \ln (r))$ .

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Step 8.3.1

Combine $5$ and $\frac{1}{3}$ .

$\frac{\ln (2)}{3} + \frac{\ln (3)}{3} + \frac{5}{3} \ln (r) + \frac{1}{3} \ln (s)$

Step 8.3.2

Combine $\frac{5}{3}$ and $\ln (r)$ .

$\frac{\ln (2)}{3} + \frac{\ln (3)}{3} + \frac{5 \ln (r)}{3} + \frac{1}{3} \ln (s)$

Step 8.4

Combine $\frac{1}{3}$ and $\ln (s)$ .

$\frac{\ln (2)}{3} + \frac{\ln (3)}{3} + \frac{5 \ln (r)}{3} + \frac{\ln (s)}{3}$