Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 8

Simplify the denominator.

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Step 8.1

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

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Step 8.1.1

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

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

Step 8.1.2

Cancel the common factor of $2$ .

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Step 8.1.2.1

Cancel the common factor.

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

Step 8.1.2.2

Rewrite the expression.

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

Step 8.2

Simplify.

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

Step 8.3

Multiply the exponents in ${(x^{3})}^{\frac{1}{2}}$ .

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Step 8.3.1

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

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

Step 8.3.2

Combine $3$ and $\frac{1}{2}$ .

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

Step 9

Reorder factors in $\log (\frac{{(x + 1)}^{\frac{1}{2}}}{(x - 1) x^{\frac{3}{2}}})$ .

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