Precalculus Examples

$\frac{2}{3} \cdot \ln (8) + \ln (x) + \ln (x^{2} - 4) - \ln (x - 2)$

Step 1

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

$\frac{2}{3} \cdot \ln (8) + \ln (x (x^{2} - 4)) - \ln (x - 2)$

Step 2

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

$\frac{2}{3} \cdot \ln (8) + \ln (\frac{x (x^{2} - 4)}{x - 2})$

Step 3

Simplify each term.

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Step 3.1

Simplify $\frac{2}{3} \ln (8)$ by moving $\frac{2}{3}$ inside the logarithm.

$\ln (8^{\frac{2}{3}}) + \ln (\frac{x (x^{2} - 4)}{x - 2})$

Step 3.2

Rewrite $8$ as $2^{3}$ .

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

Step 3.3

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

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

Step 3.4

Cancel the common factor of $3$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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

Step 3.5

Raise $2$ to the power of $2$ .

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

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

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

Step 3.6.2

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 $b = 2$ .

$\ln (4) + \ln (\frac{x (x + 2) (x - 2)}{x - 2})$

Step 3.7

Cancel the common factor of $x - 2$ .

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Step 3.7.1

Cancel the common factor.

$\ln (4) + \ln (\frac{x (x + 2) (x - 2)}{x - 2})$

Step 3.7.2

Divide $x (x + 2)$ by $1$ .

$\ln (4) + \ln (x (x + 2))$

Step 3.8

Apply the distributive property.

$\ln (4) + \ln (x \cdot x + x \cdot 2)$

Step 3.9

Multiply $x$ by $x$ .

$\ln (4) + \ln (x^{2} + x \cdot 2)$

Step 3.10

Move $2$ to the left of $x$ .

$\ln (4) + \ln (x^{2} + 2 x)$

Step 4

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

$\ln (4 (x^{2} + 2 x))$

Step 5

Apply the distributive property.

$\ln (4 x^{2} + 4 (2 x))$

Step 6

Multiply $2$ by $4$ .

$\ln (4 x^{2} + 8 x)$