Precalculus Examples

$\frac{1}{5} (3 \log (x + 1) + 2 \log (x - 1) - \log (7))$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log (x + 1)$ by moving $3$ inside the logarithm.

$\frac{1}{5} \cdot (\log ({(x + 1)}^{3}) + 2 \log (x - 1) - \log (7))$

Step 1.2

Simplify $2 \log (x - 1)$ by moving $2$ inside the logarithm.

$\frac{1}{5} \cdot (\log ({(x + 1)}^{3}) + \log ({(x - 1)}^{2}) - \log (7))$

Step 2

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

$\frac{1}{5} \cdot (\log ({(x + 1)}^{3} {(x - 1)}^{2}) - \log (7))$

Step 3

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

$\frac{1}{5} \cdot \log (\frac{{(x + 1)}^{3} {(x - 1)}^{2}}{7})$

Step 4

Simplify $\frac{1}{5} \log (\frac{{(x + 1)}^{3} {(x - 1)}^{2}}{7})$ by moving $\frac{1}{5}$ inside the logarithm.

$\log ({(\frac{{(x + 1)}^{3} {(x - 1)}^{2}}{7})}^{\frac{1}{5}})$

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 $\frac{{(x + 1)}^{3} {(x - 1)}^{2}}{7}$ .

$\log (\frac{{({(x + 1)}^{3} {(x - 1)}^{2})}^{\frac{1}{5}}}{7^{\frac{1}{5}}})$

Step 5.2

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

$\log (\frac{{({(x + 1)}^{3})}^{\frac{1}{5}} {({(x - 1)}^{2})}^{\frac{1}{5}}}{7^{\frac{1}{5}}})$

Step 6

Simplify the numerator.

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Step 6.1

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

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Step 6.1.1

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

$\log (\frac{{(x + 1)}^{3 (\frac{1}{5})} {({(x - 1)}^{2})}^{\frac{1}{5}}}{7^{\frac{1}{5}}})$

Step 6.1.2

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

$\log (\frac{{(x + 1)}^{\frac{3}{5}} {({(x - 1)}^{2})}^{\frac{1}{5}}}{7^{\frac{1}{5}}})$

Step 6.2

Multiply the exponents in ${({(x - 1)}^{2})}^{\frac{1}{5}}$ .

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Step 6.2.1

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

$\log (\frac{{(x + 1)}^{\frac{3}{5}} {(x - 1)}^{2 (\frac{1}{5})}}{7^{\frac{1}{5}}})$

Step 6.2.2

Combine $2$ and $\frac{1}{5}$ .

$\log (\frac{{(x + 1)}^{\frac{3}{5}} {(x - 1)}^{\frac{2}{5}}}{7^{\frac{1}{5}}})$