Precalculus Examples

$3 \log (x) - \log (x^{2} - 4) + 2 \log (x - 2)$

Step 1

Simplify each term.

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Step 1.1

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

$\log (x^{3}) - \log (x^{2} - 4) + 2 \log (x - 2)$

Step 1.2

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

$\log (x^{3}) - \log (x^{2} - 4) + \log ({(x - 2)}^{2})$

Step 2

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

$\log (\frac{x^{3}}{x^{2} - 4}) + \log ({(x - 2)}^{2})$

Step 3

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

$\log (\frac{x^{3}}{x^{2} - 4} {(x - 2)}^{2})$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $4$ as $2^{2}$ .

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

Step 4.2

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 $b = 2$ .

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

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x - 2$ .

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Step 5.1.1

Factor $x - 2$ out of $(x + 2) (x - 2)$ .

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

Step 5.1.2

Factor $x - 2$ out of ${(x - 2)}^{2}$ .

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

Step 5.1.3

Cancel the common factor.

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

Step 5.1.4

Rewrite the expression.

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

Step 5.2

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

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