Precalculus Examples

$\ln (11^{3} \sqrt{7 \cdot 12})$

Step 1

Rewrite $\ln (11^{3} \sqrt{7 \cdot 12})$ as $\ln (11^{3}) + \ln (\sqrt{7 \cdot 12})$ .

$\ln (11^{3}) + \ln (\sqrt{7 \cdot 12})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 \cdot 12}$ as ${(7 \cdot 12)}^{\frac{1}{2}}$ .

$\ln (11^{3}) + \ln ({(7 \cdot 12)}^{\frac{1}{2}})$

Step 3

Expand $\ln (11^{3})$ by moving $3$ outside the logarithm.

$3 \ln (11) + \ln ({(7 \cdot 12)}^{\frac{1}{2}})$

Step 4

Expand $\ln ({(7 \cdot 12)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$3 \ln (11) + \frac{1}{2} \ln (7 \cdot 12)$

Step 5

Multiply $7$ by $12$ .

$3 \ln (11) + \frac{1}{2} \ln (84)$

Step 6

Rewrite $\ln (84)$ as $\ln (2^{2} \cdot 3 \cdot 7)$ .

$3 \ln (11) + \frac{1}{2} \ln (2^{2} \cdot 3 \cdot 7)$

Step 7

Rewrite $\ln (2^{2} \cdot 3 \cdot 7)$ as $\ln (2^{2} \cdot 3) + \ln (7)$ .

$3 \ln (11) + \frac{1}{2} (\ln (2^{2} \cdot 3) + \ln (7))$

Step 8

Raise $2$ to the power of $2$ .

$3 \ln (11) + \frac{1}{2} (\ln (4 \cdot 3) + \ln (7))$

Step 9

Multiply $4$ by $3$ .

$3 \ln (11) + \frac{1}{2} (\ln (12) + \ln (7))$

Step 10

Simplify each term.

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Step 10.1

Rewrite $\ln (12)$ as $\ln (2^{2} \cdot 3)$ .

$3 \ln (11) + \frac{1}{2} (\ln (2^{2} \cdot 3) + \ln (7))$

Step 10.2

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

$3 \ln (11) + \frac{1}{2} (\ln (2^{2}) + \ln (3) + \ln (7))$

Step 10.3

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

$3 \ln (11) + \frac{1}{2} (2 \ln (2) + \ln (3) + \ln (7))$

Step 11

Apply the distributive property.

$3 \ln (11) + \frac{1}{2} (2 \ln (2)) + \frac{1}{2} \ln (3) + \frac{1}{2} \ln (7)$

Step 12

Simplify.

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Step 12.1

Cancel the common factor of $2$ .

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Step 12.1.1

Factor $2$ out of $2 \ln (2)$ .

$3 \ln (11) + \frac{1}{2} (2 (\ln (2))) + \frac{1}{2} \ln (3) + \frac{1}{2} \ln (7)$

Step 12.1.2

Cancel the common factor.

$3 \ln (11) + \frac{1}{2} (2 \ln (2)) + \frac{1}{2} \ln (3) + \frac{1}{2} \ln (7)$

Step 12.1.3

Rewrite the expression.

$3 \ln (11) + \ln (2) + \frac{1}{2} \ln (3) + \frac{1}{2} \ln (7)$

Step 12.2

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

$3 \ln (11) + \ln (2) + \frac{\ln (3)}{2} + \frac{1}{2} \ln (7)$

Step 12.3

Combine $\frac{1}{2}$ and $\ln (7)$ .

$3 \ln (11) + \ln (2) + \frac{\ln (3)}{2} + \frac{\ln (7)}{2}$