Precalculus Examples

$6 \ln (x - 4) + \frac{1}{6} \cdot \ln ({(x)}^{36}) - 12 \ln ({(x + 2)}^{9})$

Step 1

Simplify each term.

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Step 1.1

Simplify $6 \ln (x - 4)$ by moving $6$ inside the logarithm.

$\ln ({(x - 4)}^{6}) + \frac{1}{6} \cdot \ln ({(x)}^{36}) - 12 \ln ({(x + 2)}^{9})$

Step 1.2

Simplify $\frac{1}{6} \ln (x^{36})$ by moving $\frac{1}{6}$ inside the logarithm.

$\ln ({(x - 4)}^{6}) + \ln ({(x^{36})}^{\frac{1}{6}}) - 12 \ln ({(x + 2)}^{9})$

Step 1.3

Multiply the exponents in ${(x^{36})}^{\frac{1}{6}}$ .

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Step 1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\ln ({(x - 4)}^{6}) + \ln (x^{36 (\frac{1}{6})}) - 12 \ln ({(x + 2)}^{9})$

Step 1.3.2

Cancel the common factor of $6$ .

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Step 1.3.2.1

Factor $6$ out of $36$ .

$\ln ({(x - 4)}^{6}) + \ln (x^{6 (6) \frac{1}{6}}) - 12 \ln ({(x + 2)}^{9})$

Step 1.3.2.2

Cancel the common factor.

$\ln ({(x - 4)}^{6}) + \ln (x^{6 \cdot 6 \frac{1}{6}}) - 12 \ln ({(x + 2)}^{9})$

Step 1.3.2.3

Rewrite the expression.

$\ln ({(x - 4)}^{6}) + \ln (x^{6}) - 12 \ln ({(x + 2)}^{9})$

Step 1.4

Simplify $- 12 \ln ({(x + 2)}^{9})$ by moving $12$ inside the logarithm.

$\ln ({(x - 4)}^{6}) + \ln (x^{6}) - \ln ({({(x + 2)}^{9})}^{12})$

Step 1.5

Multiply the exponents in ${({(x + 2)}^{9})}^{12}$ .

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Step 1.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\ln ({(x - 4)}^{6}) + \ln (x^{6}) - \ln ({(x + 2)}^{9 \cdot 12})$

Step 1.5.2

Multiply $9$ by $12$ .

$\ln ({(x - 4)}^{6}) + \ln (x^{6}) - \ln ({(x + 2)}^{108})$

Step 2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln ({(x - 4)}^{6} x^{6}) - \ln ({(x + 2)}^{108})$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{{(x - 4)}^{6} x^{6}}{{(x + 2)}^{108}})$

Step 4

Reorder factors in $\ln (\frac{{(x - 4)}^{6} x^{6}}{{(x + 2)}^{108}})$ .

$\ln (\frac{x^{6} {(x - 4)}^{6}}{{(x + 2)}^{108}})$