Precalculus Examples

$\frac{1}{4} \cdot (2 \ln (x + 1) + 3 \ln (x - 1) - \ln (6))$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \ln (x + 1)$ by moving $2$ inside the logarithm.

$\frac{1}{4} \cdot (\ln ({(x + 1)}^{2}) + 3 \ln (x - 1) - \ln (6))$

Step 1.2

Simplify $3 \ln (x - 1)$ by moving $3$ inside the logarithm.

$\frac{1}{4} \cdot (\ln ({(x + 1)}^{2}) + \ln ({(x - 1)}^{3}) - \ln (6))$

Step 2

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

$\frac{1}{4} \cdot (\ln ({(x + 1)}^{2} {(x - 1)}^{3}) - \ln (6))$

Step 3

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

$\frac{1}{4} \cdot \ln (\frac{{(x + 1)}^{2} {(x - 1)}^{3}}{6})$

Step 4

Simplify $\frac{1}{4} \ln (\frac{{(x + 1)}^{2} {(x - 1)}^{3}}{6})$ by moving $\frac{1}{4}$ inside the logarithm.

$\ln ({(\frac{{(x + 1)}^{2} {(x - 1)}^{3}}{6})}^{\frac{1}{4}})$

Step 5

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

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Step 5.1

Apply the product rule to $\frac{{(x + 1)}^{2} {(x - 1)}^{3}}{6}$ .

$\ln (\frac{{({(x + 1)}^{2} {(x - 1)}^{3})}^{\frac{1}{4}}}{6^{\frac{1}{4}}})$

Step 5.2

Apply the product rule to ${(x + 1)}^{2} {(x - 1)}^{3}$ .

$\ln (\frac{{({(x + 1)}^{2})}^{\frac{1}{4}} {({(x - 1)}^{3})}^{\frac{1}{4}}}{6^{\frac{1}{4}}})$

Step 6

Simplify the numerator.

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Step 6.1

Multiply the exponents in ${({(x + 1)}^{2})}^{\frac{1}{4}}$ .

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Step 6.1.1

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

$\ln (\frac{{(x + 1)}^{2 (\frac{1}{4})} {({(x - 1)}^{3})}^{\frac{1}{4}}}{6^{\frac{1}{4}}})$

Step 6.1.2

Cancel the common factor of $2$ .

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Step 6.1.2.1

Factor $2$ out of $4$ .

$\ln (\frac{{(x + 1)}^{2 \frac{1}{2 (2)}} {({(x - 1)}^{3})}^{\frac{1}{4}}}{6^{\frac{1}{4}}})$

Step 6.1.2.2

Cancel the common factor.

$\ln (\frac{{(x + 1)}^{2 \frac{1}{2 \cdot 2}} {({(x - 1)}^{3})}^{\frac{1}{4}}}{6^{\frac{1}{4}}})$

Step 6.1.2.3

Rewrite the expression.

$\ln (\frac{{(x + 1)}^{\frac{1}{2}} {({(x - 1)}^{3})}^{\frac{1}{4}}}{6^{\frac{1}{4}}})$

Step 6.2

Multiply the exponents in ${({(x - 1)}^{3})}^{\frac{1}{4}}$ .

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Step 6.2.1

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

$\ln (\frac{{(x + 1)}^{\frac{1}{2}} {(x - 1)}^{3 (\frac{1}{4})}}{6^{\frac{1}{4}}})$

Step 6.2.2

Combine $3$ and $\frac{1}{4}$ .

$\ln (\frac{{(x + 1)}^{\frac{1}{2}} {(x - 1)}^{\frac{3}{4}}}{6^{\frac{1}{4}}})$