Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Raise $3$ to the power of $2$ .

$\frac{1}{2} \cdot (\log (4) + \log (9) - (\log (2) + \log (3)))$

Step 1.3

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

$\frac{1}{2} \cdot (\log (4) + \log (9) - \log (2 \cdot 3))$

Step 1.4

Multiply $2$ by $3$ .

$\frac{1}{2} \cdot (\log (4) + \log (9) - \log (6))$

Step 2

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

$\frac{1}{2} \cdot (\log (4 \cdot 9) - \log (6))$

Step 3

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

$\frac{1}{2} \cdot \log (\frac{4 \cdot 9}{6})$

Step 4

Cancel the common factor of $4$ and $6$ .

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Step 4.1

Factor $2$ out of $4 \cdot 9$ .

$\frac{1}{2} \cdot \log (\frac{2 (2 \cdot 9)}{6})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $2$ out of $6$ .

$\frac{1}{2} \cdot \log (\frac{2 (2 \cdot 9)}{2 (3)})$

Step 4.2.2

Cancel the common factor.

$\frac{1}{2} \cdot \log (\frac{2 (2 \cdot 9)}{2 \cdot 3})$

Step 4.2.3

Rewrite the expression.

$\frac{1}{2} \cdot \log (\frac{2 \cdot 9}{3})$

Step 5

Cancel the common factor of $9$ and $3$ .

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Step 5.1

Factor $3$ out of $2 \cdot 9$ .

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $3$ out of $3$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 5.2.4

Divide $2 \cdot 3$ by $1$ .

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

Step 6

Multiply $2$ by $3$ .

$\frac{1}{2} \cdot \log (6)$

Step 7

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

$\log (6^{\frac{1}{2}})$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\log (6^{\frac{1}{2}})$

Decimal Form:

$0.38907562 \dots$