Precalculus Examples

$\frac{1}{3} \cdot (\ln (x - 2) + 2 \ln (x + 2) - \ln (x + 2) - 5 \ln (x - 1))$

Step 1

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

$\frac{1}{3} \cdot (2 \ln (x + 2) + \ln (\frac{x - 2}{x + 2}) - 5 \ln (x - 1))$

Step 2

Simplify each term.

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Step 2.1

Simplify $2 \ln (x + 2)$ by moving $2$ inside the logarithm.

$\frac{1}{3} \cdot (\ln ({(x + 2)}^{2}) + \ln (\frac{x - 2}{x + 2}) - 5 \ln (x - 1))$

Step 2.2

Simplify $- 5 \ln (x - 1)$ by moving $5$ inside the logarithm.

$\frac{1}{3} \cdot (\ln ({(x + 2)}^{2}) + \ln (\frac{x - 2}{x + 2}) - \ln ({(x - 1)}^{5}))$

Step 3

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

$\frac{1}{3} \cdot (\ln ({(x + 2)}^{2} \frac{x - 2}{x + 2}) - \ln ({(x - 1)}^{5}))$

Step 4

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

$\frac{1}{3} \cdot \ln (\frac{{(x + 2)}^{2} \frac{x - 2}{x + 2}}{{(x - 1)}^{5}})$

Step 5

Combine ${(x + 2)}^{2}$ and $\frac{x - 2}{x + 2}$ .

$\frac{1}{3} \cdot \ln (\frac{\frac{{(x + 2)}^{2} (x - 2)}{x + 2}}{{(x - 1)}^{5}})$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Reduce the expression $\frac{{(x + 2)}^{2} (x - 2)}{x + 2}$ by cancelling the common factors.

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Step 6.1.1

Factor $x + 2$ out of ${(x + 2)}^{2} (x - 2)$ .

$\frac{1}{3} \cdot \ln (\frac{\frac{(x + 2) ((x + 2) (x - 2))}{x + 2}}{{(x - 1)}^{5}})$

Step 6.1.2

Multiply by $1$ .

$\frac{1}{3} \cdot \ln (\frac{\frac{(x + 2) ((x + 2) (x - 2))}{(x + 2) \cdot 1}}{{(x - 1)}^{5}})$

Step 6.1.3

Cancel the common factor.

$\frac{1}{3} \cdot \ln (\frac{\frac{(x + 2) ((x + 2) (x - 2))}{(x + 2) \cdot 1}}{{(x - 1)}^{5}})$

Step 6.1.4

Rewrite the expression.

$\frac{1}{3} \cdot \ln (\frac{\frac{(x + 2) (x - 2)}{1}}{{(x - 1)}^{5}})$

Step 6.2

Divide $(x + 2) (x - 2)$ by $1$ .

$\frac{1}{3} \cdot \ln (\frac{(x + 2) (x - 2)}{{(x - 1)}^{5}})$

Step 7

Simplify $\frac{1}{3} \ln (\frac{(x + 2) (x - 2)}{{(x - 1)}^{5}})$ by moving $\frac{1}{3}$ inside the logarithm.

$\ln ({(\frac{(x + 2) (x - 2)}{{(x - 1)}^{5}})}^{\frac{1}{3}})$

Step 8

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.1

Apply the product rule to $\frac{(x + 2) (x - 2)}{{(x - 1)}^{5}}$ .

$\ln (\frac{{((x + 2) (x - 2))}^{\frac{1}{3}}}{{({(x - 1)}^{5})}^{\frac{1}{3}}})$

Step 8.2

Apply the product rule to $(x + 2) (x - 2)$ .

$\ln (\frac{{(x + 2)}^{\frac{1}{3}} {(x - 2)}^{\frac{1}{3}}}{{({(x - 1)}^{5})}^{\frac{1}{3}}})$

Step 9

Multiply the exponents in ${({(x - 1)}^{5})}^{\frac{1}{3}}$ .

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Step 9.1

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

$\ln (\frac{{(x + 2)}^{\frac{1}{3}} {(x - 2)}^{\frac{1}{3}}}{{(x - 1)}^{5 (\frac{1}{3})}})$

Step 9.2

Combine $5$ and $\frac{1}{3}$ .

$\ln (\frac{{(x + 2)}^{\frac{1}{3}} {(x - 2)}^{\frac{1}{3}}}{{(x - 1)}^{\frac{5}{3}}})$