Precalculus Examples

$\ln ({(\frac{(x + 6) (x - 2)}{{(x - 8)}^{2}})}^{\frac{3}{4}})$

Step 1

Expand $\ln ({(\frac{(x + 6) (x - 2)}{{(x - 8)}^{2}})}^{\frac{3}{4}})$ by moving $\frac{3}{4}$ outside the logarithm.

$\frac{3}{4} \ln (\frac{(x + 6) (x - 2)}{{(x - 8)}^{2}})$

Step 2

Rewrite $\ln (\frac{(x + 6) (x - 2)}{{(x - 8)}^{2}})$ as $\ln ((x + 6) (x - 2)) - \ln ({(x - 8)}^{2})$ .

$\frac{3}{4} (\ln ((x + 6) (x - 2)) - \ln ({(x - 8)}^{2}))$

Step 3

Expand $\ln ({(x - 8)}^{2})$ by moving $2$ outside the logarithm.

$\frac{3}{4} (\ln ((x + 6) (x - 2)) - (2 \ln (x - 8)))$

Step 4

Multiply $2$ by $- 1$ .

$\frac{3}{4} (\ln ((x + 6) (x - 2)) - 2 \ln (x - 8))$

Step 5

Rewrite $\ln ((x + 6) (x - 2))$ as $\ln (x + 6) + \ln (x - 2)$ .

$\frac{3}{4} (\ln (x + 6) + \ln (x - 2) - 2 \ln (x - 8))$

Step 6

Apply the distributive property.

$\frac{3}{4} \ln (x + 6) + \frac{3}{4} \ln (x - 2) + \frac{3}{4} (- 2 \ln (x - 8))$

Step 7

Simplify.

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Step 7.1

Combine $\frac{3}{4}$ and $\ln (x + 6)$ .

$\frac{3 \ln (x + 6)}{4} + \frac{3}{4} \ln (x - 2) + \frac{3}{4} (- 2 \ln (x - 8))$

Step 7.2

Combine $\frac{3}{4}$ and $\ln (x - 2)$ .

$\frac{3 \ln (x + 6)}{4} + \frac{3 \ln (x - 2)}{4} + \frac{3}{4} (- 2 \ln (x - 8))$

Step 7.3

Cancel the common factor of $2$ .

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Step 7.3.1

Factor $2$ out of $4$ .

$\frac{3 \ln (x + 6)}{4} + \frac{3 \ln (x - 2)}{4} + \frac{3}{2 (2)} (- 2 \ln (x - 8))$

Step 7.3.2

Factor $2$ out of $- 2 \ln (x - 8)$ .

$\frac{3 \ln (x + 6)}{4} + \frac{3 \ln (x - 2)}{4} + \frac{3}{2 (2)} (2 (- \ln (x - 8)))$

Step 7.3.3

Cancel the common factor.

$\frac{3 \ln (x + 6)}{4} + \frac{3 \ln (x - 2)}{4} + \frac{3}{2 \cdot 2} (2 (- \ln (x - 8)))$

Step 7.3.4

Rewrite the expression.

$\frac{3 \ln (x + 6)}{4} + \frac{3 \ln (x - 2)}{4} + \frac{3}{2} (- \ln (x - 8))$

Step 7.4

Combine $\frac{3}{2}$ and $\ln (x - 8)$ .

$\frac{3 \ln (x + 6)}{4} + \frac{3 \ln (x - 2)}{4} - \frac{3 \ln (x - 8)}{2}$

Step 8

Combine the numerators over the common denominator.

$\frac{3 \ln (x + 6) + 3 \ln (x - 2)}{4} - \frac{3 \ln (x - 8)}{2}$

Step 9

Factor $3$ out of $3 \ln (x + 6) + 3 \ln (x - 2)$ .

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Step 9.1

Factor $3$ out of $3 \ln (x + 6)$ .

$\frac{3 (\ln (x + 6)) + 3 \ln (x - 2)}{4} - \frac{3 \ln (x - 8)}{2}$

Step 9.2

Factor $3$ out of $3 \ln (x - 2)$ .

$\frac{3 (\ln (x + 6)) + 3 (\ln (x - 2))}{4} - \frac{3 \ln (x - 8)}{2}$

Step 9.3

Factor $3$ out of $3 (\ln (x + 6)) + 3 (\ln (x - 2))$ .

$\frac{3 (\ln (x + 6) + \ln (x - 2))}{4} - \frac{3 \ln (x - 8)}{2}$

Step 10

To write $- \frac{3 \ln (x - 8)}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (\ln (x + 6) + \ln (x - 2))}{4} - \frac{3 \ln (x - 8)}{2} \cdot \frac{2}{2}$

Step 11

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.1

Multiply $\frac{3 \ln (x - 8)}{2}$ by $\frac{2}{2}$ .

$\frac{3 (\ln (x + 6) + \ln (x - 2))}{4} - \frac{3 \ln (x - 8) \cdot 2}{2 \cdot 2}$

Step 11.2

Multiply $2$ by $2$ .

$\frac{3 (\ln (x + 6) + \ln (x - 2))}{4} - \frac{3 \ln (x - 8) \cdot 2}{4}$

Step 12

Combine the numerators over the common denominator.

$\frac{3 (\ln (x + 6) + \ln (x - 2)) - 3 \ln (x - 8) \cdot 2}{4}$

Step 13

Simplify the numerator.

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Step 13.1

Factor $3$ out of $3 (\ln (x + 6) + \ln (x - 2)) - 3 \ln (x - 8) \cdot 2$ .

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Step 13.1.1

Factor $3$ out of $- 3 \ln (x - 8) \cdot 2$ .

$\frac{3 (\ln (x + 6) + \ln (x - 2)) + 3 (- \ln (x - 8) \cdot 2)}{4}$

Step 13.1.2

Factor $3$ out of $3 (\ln (x + 6) + \ln (x - 2)) + 3 (- \ln (x - 8) \cdot 2)$ .

$\frac{3 (\ln (x + 6) + \ln (x - 2) - \ln (x - 8) \cdot 2)}{4}$

Step 13.2

Multiply $2$ by $- 1$ .

$\frac{3 (\ln (x + 6) + \ln (x - 2) - 2 \ln (x - 8))}{4}$