Precalculus Examples

$4 \log ({(\frac{1}{9} \cdot 3)}^{2}) - \log (\frac{1}{9}) \cdot 3 + 5$

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $3$ .

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Step 1.1.1

Factor $3$ out of $9$ .

$4 \log ({(\frac{1}{3 (3)} \cdot 3)}^{2}) - \log (\frac{1}{9}) \cdot 3 + 5$

Step 1.1.2

Cancel the common factor.

$4 \log ({(\frac{1}{3 \cdot 3} \cdot 3)}^{2}) - \log (\frac{1}{9}) \cdot 3 + 5$

Step 1.1.3

Rewrite the expression.

$4 \log ({(\frac{1}{3})}^{2}) - \log (\frac{1}{9}) \cdot 3 + 5$

Step 1.2

Apply the product rule to $\frac{1}{3}$ .

$4 \log (\frac{1^{2}}{3^{2}}) - \log (\frac{1}{9}) \cdot 3 + 5$

Step 1.3

One to any power is one.

$4 \log (\frac{1}{3^{2}}) - \log (\frac{1}{9}) \cdot 3 + 5$

Step 1.4

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

$4 \log (\frac{1}{9}) - \log (\frac{1}{9}) \cdot 3 + 5$

Step 1.5

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

$\log ({(\frac{1}{9})}^{4}) - \log (\frac{1}{9}) \cdot 3 + 5$

Step 1.6

Apply the product rule to $\frac{1}{9}$ .

$\log (\frac{1^{4}}{9^{4}}) - \log (\frac{1}{9}) \cdot 3 + 5$

Step 1.7

One to any power is one.

$\log (\frac{1}{9^{4}}) - \log (\frac{1}{9}) \cdot 3 + 5$

Step 1.8

Raise $9$ to the power of $4$ .

$\log (\frac{1}{6561}) - \log (\frac{1}{9}) \cdot 3 + 5$

Step 1.9

Multiply $- \log (\frac{1}{9}) \cdot 3$ .

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Step 1.9.1

Multiply $3$ by $- 1$ .

$\log (\frac{1}{6561}) - 3 \log (\frac{1}{9}) + 5$

Step 1.9.2

Simplify $- 3 \log (\frac{1}{9})$ by moving $3$ inside the logarithm.

$\log (\frac{1}{6561}) - \log ({(\frac{1}{9})}^{3}) + 5$

Step 1.10

Apply the product rule to $\frac{1}{9}$ .

$\log (\frac{1}{6561}) - \log (\frac{1^{3}}{9^{3}}) + 5$

Step 1.11

One to any power is one.

$\log (\frac{1}{6561}) - \log (\frac{1}{9^{3}}) + 5$

Step 1.12

Raise $9$ to the power of $3$ .

$\log (\frac{1}{6561}) - \log (\frac{1}{729}) + 5$

Step 2

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

$\log (\frac{\frac{1}{6561}}{\frac{1}{729}}) + 5$

Step 3

Simplify each term.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{1}{6561} \cdot 729) + 5$

Step 3.2

Cancel the common factor of $729$ .

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Step 3.2.1

Factor $729$ out of $6561$ .

$\log (\frac{1}{729 (9)} \cdot 729) + 5$

Step 3.2.2

Cancel the common factor.

$\log (\frac{1}{729 \cdot 9} \cdot 729) + 5$

Step 3.2.3

Rewrite the expression.

$\log (\frac{1}{9}) + 5$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\log (\frac{1}{9}) + 5$

Decimal Form:

$4.04575749 \dots$