Precalculus Examples

$\log (x) - \log (x^{2} - 81) - \log (4) - \log (x - 9)$

Step 1

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

$\log (\frac{x}{x^{2} - 81}) - \log (4) - \log (x - 9)$

Step 2

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

$\log (\frac{\frac{x}{x^{2} - 81}}{4}) - \log (x - 9)$

Step 3

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

$\log (\frac{\frac{\frac{x}{x^{2} - 81}}{4}}{x - 9})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{\frac{x}{x^{2} - 81}}{4} \cdot \frac{1}{x - 9})$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{x}{x^{2} - 81} \cdot \frac{1}{4} \frac{1}{x - 9})$

Step 6

Simplify the denominator.

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Step 6.1

Rewrite $81$ as $9^{2}$ .

$\log (\frac{x}{x^{2} - 9^{2}} \cdot \frac{1}{4} \frac{1}{x - 9})$

Step 6.2

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 $b = 9$ .

$\log (\frac{x}{(x + 9) (x - 9)} \cdot \frac{1}{4} \frac{1}{x - 9})$

Step 7

Combine fractions.

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Step 7.1

Multiply $\frac{x}{(x + 9) (x - 9)}$ by $\frac{1}{4}$ .

$\log (\frac{x}{(x + 9) (x - 9) \cdot 4} \cdot \frac{1}{x - 9})$

Step 7.2

Move $4$ to the left of $(x + 9) (x - 9)$ .

$\log (\frac{x}{4 \cdot ((x + 9) (x - 9))} \cdot \frac{1}{x - 9})$

Step 8

Multiply $\frac{x}{4 (x + 9) (x - 9)} \cdot \frac{1}{x - 9}$ .

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Step 8.1

Multiply $\frac{x}{4 (x + 9) (x - 9)}$ by $\frac{1}{x - 9}$ .

$\log (\frac{x}{4 (x + 9) (x - 9) (x - 9)})$

Step 8.2

Raise $x - 9$ to the power of $1$ .

$\log (\frac{x}{4 (x + 9) ({(x - 9)}^{1} (x - 9))})$

Step 8.3

Raise $x - 9$ to the power of $1$ .

$\log (\frac{x}{4 (x + 9) ({(x - 9)}^{1} {(x - 9)}^{1})})$

Step 8.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log (\frac{x}{4 (x + 9) {(x - 9)}^{1 + 1}})$

Step 8.5

Add $1$ and $1$ .

$\log (\frac{x}{4 (x + 9) {(x - 9)}^{2}})$