Precalculus Examples

$\frac{1}{5} \cdot (5 \ln (x + 9) - \ln (x) - \ln (x^{2} - 9))$

Step 1

Simplify $5 \ln (x + 9)$ by moving $5$ inside the logarithm.

$\frac{1}{5} \cdot (\ln ({(x + 9)}^{5}) - \ln (x) - \ln (x^{2} - 9))$

Step 2

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

$\frac{1}{5} \cdot (\ln (\frac{{(x + 9)}^{5}}{x}) - \ln (x^{2} - 9))$

Step 3

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

$\frac{1}{5} \cdot \ln (\frac{\frac{{(x + 9)}^{5}}{x}}{x^{2} - 9})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{5} \cdot \ln (\frac{{(x + 9)}^{5}}{x} \cdot \frac{1}{x^{2} - 9})$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $9$ as $3^{2}$ .

$\frac{1}{5} \cdot \ln (\frac{{(x + 9)}^{5}}{x} \cdot \frac{1}{x^{2} - 3^{2}})$

Step 5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$\frac{1}{5} \cdot \ln (\frac{{(x + 9)}^{5}}{x} \cdot \frac{1}{(x + 3) (x - 3)})$

Step 6

Combine fractions.

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Step 6.1

Combine.

$\frac{1}{5} \cdot \ln (\frac{{(x + 9)}^{5} \cdot 1}{x ((x + 3) (x - 3))})$

Step 6.2

Multiply ${(x + 9)}^{5}$ by $1$ .

$\frac{1}{5} \cdot \ln (\frac{{(x + 9)}^{5}}{x ((x + 3) (x - 3))})$

Step 7

Simplify $\frac{1}{5} \ln (\frac{{(x + 9)}^{5}}{x (x + 3) (x - 3)})$ by moving $\frac{1}{5}$ inside the logarithm.

$\ln ({(\frac{{(x + 9)}^{5}}{x (x + 3) (x - 3)})}^{\frac{1}{5}})$

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 $\frac{{(x + 9)}^{5}}{x (x + 3) (x - 3)}$ .

$\ln (\frac{{({(x + 9)}^{5})}^{\frac{1}{5}}}{{(x (x + 3) (x - 3))}^{\frac{1}{5}}})$

Step 8.2

Apply the product rule to $x (x + 3) (x - 3)$ .

$\ln (\frac{{({(x + 9)}^{5})}^{\frac{1}{5}}}{{(x (x + 3))}^{\frac{1}{5}} {(x - 3)}^{\frac{1}{5}}})$

Step 8.3

Apply the product rule to $x (x + 3)$ .

$\ln (\frac{{({(x + 9)}^{5})}^{\frac{1}{5}}}{x^{\frac{1}{5}} {(x + 3)}^{\frac{1}{5}} {(x - 3)}^{\frac{1}{5}}})$

Step 9

Simplify the numerator.

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Step 9.1

Multiply the exponents in ${({(x + 9)}^{5})}^{\frac{1}{5}}$ .

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Step 9.1.1

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

$\ln (\frac{{(x + 9)}^{5 (\frac{1}{5})}}{x^{\frac{1}{5}} {(x + 3)}^{\frac{1}{5}} {(x - 3)}^{\frac{1}{5}}})$

Step 9.1.2

Cancel the common factor of $5$ .

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Step 9.1.2.1

Cancel the common factor.

$\ln (\frac{{(x + 9)}^{5 (\frac{1}{5})}}{x^{\frac{1}{5}} {(x + 3)}^{\frac{1}{5}} {(x - 3)}^{\frac{1}{5}}})$

Step 9.1.2.2

Rewrite the expression.

$\ln (\frac{{(x + 9)}^{1}}{x^{\frac{1}{5}} {(x + 3)}^{\frac{1}{5}} {(x - 3)}^{\frac{1}{5}}})$

Step 9.2

Simplify.

$\ln (\frac{x + 9}{x^{\frac{1}{5}} {(x + 3)}^{\frac{1}{5}} {(x - 3)}^{\frac{1}{5}}})$