Precalculus Examples

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

Step 1

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

$3 - \frac{1}{2} \cdot (\log (6 \cdot 3) - 3 \log (2))$

Step 2

Simplify each term.

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Multiply $6$ by $3$ .

$3 - \frac{1}{2} \cdot (\log (18) - 3 \log (2))$

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

$3 - \frac{1}{2} \cdot (\log (18) - \log (2^{3}))$

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

$3 - \frac{1}{2} \cdot (\log (18) - \log (8))$

Step 3

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

$3 - \frac{1}{2} \cdot \log (\frac{18}{8})$

Step 4

Cancel the common factor of $18$ and $8$ .

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Factor $2$ out of $18$ .

$3 - \frac{1}{2} \cdot \log (\frac{2 (9)}{8})$

Cancel the common factors.

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Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Multiply $- \frac{1}{2} \log (\frac{9}{4})$ .

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Reorder $\log (\frac{9}{4})$ and $\frac{1}{2}$ .

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

Simplify $\frac{1}{2} \log (\frac{9}{4})$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 6

Apply the product rule to $\frac{9}{4}$ .

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

Step 7

Simplify the numerator.

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Rewrite $9$ as $3^{2}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 8

Simplify the denominator.

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Rewrite $4$ as $2^{2}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$3 - \log (\frac{3}{2})$

Step 9

The result can be shown in multiple forms.

Exact Form:

$3 - \log (\frac{3}{2})$

Decimal Form:

$2.82390874 \dots$