Precalculus Examples

$\ln (a^{5} \sqrt{b c^{5}})$

Step 1

Rewrite $\ln (a^{5} \sqrt{b c^{5}})$ as $\ln (a^{5}) + \ln (\sqrt{b c^{5}})$ .

$\ln (a^{5}) + \ln (\sqrt{b c^{5}})$

Step 2

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

$\ln (a^{5}) + \ln ({(b c^{5})}^{\frac{1}{2}})$

Step 3

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

$5 \ln (a) + \ln ({(b c^{5})}^{\frac{1}{2}})$

Step 4

Expand $\ln ({(b c^{5})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$5 \ln (a) + \frac{1}{2} \ln (b c^{5})$

Step 5

Rewrite $\ln (b c^{5})$ as $\ln (b) + \ln (c^{5})$ .

$5 \ln (a) + \frac{1}{2} (\ln (b) + \ln (c^{5}))$

Step 6

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

$5 \ln (a) + \frac{1}{2} (\ln (b) + 5 \ln (c))$

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

$5 \ln (a) + \frac{1}{2} \ln (b) + \frac{1}{2} (5 \ln (c))$

Step 7.2

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

$5 \ln (a) + \frac{\ln (b)}{2} + \frac{1}{2} (5 \ln (c))$

Step 7.3

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

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Step 7.3.1

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

$5 \ln (a) + \frac{\ln (b)}{2} + \frac{5}{2} \ln (c)$

Step 7.3.2

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

$5 \ln (a) + \frac{\ln (b)}{2} + \frac{5 \ln (c)}{2}$