Precalculus Examples

$5 \ln (x - 2) - \ln (x + 2) - 3 \ln (x)$

Step 1

Simplify each term.

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Step 1.1

Simplify $5 \ln (x - 2)$ by moving $5$ inside the logarithm.

$\ln ({(x - 2)}^{5}) - \ln (x + 2) - 3 \ln (x)$

Step 1.2

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

$\ln ({(x - 2)}^{5}) - \ln (x + 2) - \ln (x^{3})$

Step 2

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

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

Step 3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Multiply $\frac{{(x - 2)}^{5}}{x + 2}$ by $\frac{1}{x^{3}}$ .

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

Step 6

Reorder factors in $\ln (\frac{{(x - 2)}^{5}}{(x + 2) x^{3}})$ .

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