Precalculus Examples

$2 \log (x + 3) + \frac{1}{4} \log (x - 5) - \log (x + 2) - \log (x + 1)$

Step 1

Simplify each term.

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Step 1.1

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

$\log ({(x + 3)}^{2}) + \frac{1}{4} \cdot \log (x - 5) - \log (x + 2) - \log (x + 1)$

Step 1.2

Simplify $\frac{1}{4} \log (x - 5)$ by moving $\frac{1}{4}$ inside the logarithm.

$\log ({(x + 3)}^{2}) + \log ({(x - 5)}^{\frac{1}{4}}) - \log (x + 2) - \log (x + 1)$

Step 2

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

$\log ({(x + 3)}^{2} {(x - 5)}^{\frac{1}{4}}) - \log (x + 2) - \log (x + 1)$

Step 3

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

$\log (\frac{{(x + 3)}^{2} {(x - 5)}^{\frac{1}{4}}}{x + 2}) - \log (x + 1)$

Step 4

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

$\log (\frac{\frac{{(x + 3)}^{2} {(x - 5)}^{\frac{1}{4}}}{x + 2}}{x + 1})$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{{(x + 3)}^{2} {(x - 5)}^{\frac{1}{4}}}{x + 2} \cdot \frac{1}{x + 1})$

Step 6

Multiply $\frac{{(x + 3)}^{2} {(x - 5)}^{\frac{1}{4}}}{x + 2}$ by $\frac{1}{x + 1}$ .

$\log (\frac{{(x + 3)}^{2} {(x - 5)}^{\frac{1}{4}}}{(x + 2) (x + 1)})$

Step 7

Reorder factors in $\log (\frac{{(x + 3)}^{2} {(x - 5)}^{\frac{1}{4}}}{(x + 2) (x + 1)})$ .

$\log (\frac{{(x - 5)}^{\frac{1}{4}} {(x + 3)}^{2}}{(x + 2) (x + 1)})$