Precalculus Examples

$3 \log (x - 2) - \log (x + 1) - \frac{1}{2} \cdot \log (x - 3)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log (x - 2)$ by moving $3$ inside the logarithm.

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

Step 1.2

Multiply $- \frac{1}{2} \log (x - 3)$ .

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Step 1.2.1

Reorder $\log (x - 3)$ and $\frac{1}{2}$ .

$\log ({(x - 2)}^{3}) - \log (x + 1) - (\frac{1}{2} \log (x - 3))$

Step 1.2.2

Simplify $\frac{1}{2} \log (x - 3)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log ({(x - 2)}^{3}) - \log (x + 1) - \log ({(x - 3)}^{\frac{1}{2}})$

Step 2

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

$\log (\frac{{(x - 2)}^{3}}{x + 1}) - \log ({(x - 3)}^{\frac{1}{2}})$

Step 3

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

$\log (\frac{\frac{{(x - 2)}^{3}}{x + 1}}{{(x - 3)}^{\frac{1}{2}}})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

$\log (\frac{{(x - 2)}^{3}}{(x + 1) {(x - 3)}^{\frac{1}{2}}})$

Step 6

Reorder factors in $\log (\frac{{(x - 2)}^{3}}{(x + 1) {(x - 3)}^{\frac{1}{2}}})$ .

$\log (\frac{{(x - 2)}^{3}}{{(x - 3)}^{\frac{1}{2}} (x + 1)})$