Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Simplify $- 5 \log (2 x + 3)$ by moving $5$ inside the logarithm.

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

Step 2

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

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

Step 3

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

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