Precalculus Examples

$\ln (\frac{3 \sqrt{c}}{a^{3} b^{2}})$

Step 1

Rewrite $\ln (\frac{3 \sqrt{c}}{a^{3} b^{2}})$ as $\ln (3 \sqrt{c}) - \ln (a^{3} b^{2})$ .

$\ln (3 \sqrt{c}) - \ln (a^{3} b^{2})$

Step 2

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

$\ln (3 c^{\frac{1}{2}}) - \ln (a^{3} b^{2})$

Step 3

Rewrite $\ln (3 c^{\frac{1}{2}})$ as $\ln (3) + \ln (c^{\frac{1}{2}})$ .

$\ln (3) + \ln (c^{\frac{1}{2}}) - \ln (a^{3} b^{2})$

Step 4

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

$\ln (3) + \frac{1}{2} \ln (c) - \ln (a^{3} b^{2})$

Step 5

Rewrite $\ln (a^{3} b^{2})$ as $\ln (a^{3}) + \ln (b^{2})$ .

$\ln (3) + \frac{1}{2} \ln (c) - (\ln (a^{3}) + \ln (b^{2}))$

Step 6

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

$\ln (3) + \frac{1}{2} \ln (c) - (3 \ln (a) + \ln (b^{2}))$

Step 7

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

$\ln (3) + \frac{1}{2} \ln (c) - (3 \ln (a) + 2 \ln (b))$

Step 8

Simplify each term.

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Step 8.1

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

$\ln (3) + \frac{\ln (c)}{2} - (3 \ln (a) + 2 \ln (b))$

Step 8.2

Apply the distributive property.

$\ln (3) + \frac{\ln (c)}{2} - (3 \ln (a)) - (2 \ln (b))$

Step 8.3

Multiply $3$ by $- 1$ .

$\ln (3) + \frac{\ln (c)}{2} - 3 \ln (a) - (2 \ln (b))$

Step 8.4

Multiply $2$ by $- 1$ .

$\ln (3) + \frac{\ln (c)}{2} - 3 \ln (a) - 2 \ln (b)$