Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Expand $(x + 3) (x - 2)$ using the FOIL Method.

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Apply the distributive property.

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

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $x$ by $x$ .

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

Step 1.6.1.2

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

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

Step 1.6.1.3

Multiply $3$ by $- 2$ .

$\log (4) + \log (x^{3}) - \frac{1}{2} \cdot \log (x^{2} - 2 x + 3 x - 6)$

Step 1.6.2

Add $- 2 x$ and $3 x$ .

$\log (4) + \log (x^{3}) - \frac{1}{2} \cdot \log (x^{2} + x - 6)$

Step 1.7

Multiply $- \frac{1}{2} \log (x^{2} + x - 6)$ .

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Step 1.7.1

Reorder $\log (x^{2} + x - 6)$ and $\frac{1}{2}$ .

$\log (4) + \log (x^{3}) - (\frac{1}{2} \log (x^{2} + x - 6))$

Step 1.7.2

Simplify $\frac{1}{2} \log (x^{2} + x - 6)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log (4) + \log (x^{3}) - \log ({(x^{2} + x - 6)}^{\frac{1}{2}})$

Step 2

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

$\log (4 x^{3}) - \log ({(x^{2} + x - 6)}^{\frac{1}{2}})$

Step 3

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

$\log (\frac{4 x^{3}}{{(x^{2} + x - 6)}^{\frac{1}{2}}})$