Precalculus Examples

$2 \log_{3} (6) - \frac{1}{2} \cdot \log_{3} (400) + 3 \log_{3} (45^{\frac{1}{3}})$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{3} (6)$ by moving $2$ inside the logarithm.

$\log_{3} (6^{2}) - \frac{1}{2} \cdot \log_{3} (400) + 3 \log_{3} (45^{\frac{1}{3}})$

Step 1.2

Raise $6$ to the power of $2$ .

$\log_{3} (36) - \frac{1}{2} \cdot \log_{3} (400) + 3 \log_{3} (45^{\frac{1}{3}})$

Step 1.3

Multiply $- \frac{1}{2} \log_{3} (400)$ .

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Step 1.3.1

Reorder $\log_{3} (400)$ and $\frac{1}{2}$ .

$\log_{3} (36) - (\frac{1}{2} \log_{3} (400)) + 3 \log_{3} (45^{\frac{1}{3}})$

Step 1.3.2

Simplify $\frac{1}{2} \log_{3} (400)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{3} (36) - \log_{3} (400^{\frac{1}{2}}) + 3 \log_{3} (45^{\frac{1}{3}})$

Step 1.4

Rewrite $400$ as $20^{2}$ .

$\log_{3} (36) - \log_{3} ({(20^{2})}^{\frac{1}{2}}) + 3 \log_{3} (45^{\frac{1}{3}})$

Step 1.5

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

$\log_{3} (36) - \log_{3} (20^{2 (\frac{1}{2})}) + 3 \log_{3} (45^{\frac{1}{3}})$

Step 1.6

Cancel the common factor of $2$ .

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Step 1.6.1

Cancel the common factor.

$\log_{3} (36) - \log_{3} (20^{2 (\frac{1}{2})}) + 3 \log_{3} (45^{\frac{1}{3}})$

Step 1.6.2

Rewrite the expression.

$\log_{3} (36) - \log_{3} (20^{1}) + 3 \log_{3} (45^{\frac{1}{3}})$

Step 1.7

Evaluate the exponent.

$\log_{3} (36) - \log_{3} (20) + 3 \log_{3} (45^{\frac{1}{3}})$

Step 1.8

Simplify $3 \log_{3} (45^{\frac{1}{3}})$ by moving $3$ inside the logarithm.

$\log_{3} (36) - \log_{3} (20) + \log_{3} ({(45^{\frac{1}{3}})}^{3})$

Step 1.9

Multiply the exponents in ${(45^{\frac{1}{3}})}^{3}$ .

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Step 1.9.1

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

$\log_{3} (36) - \log_{3} (20) + \log_{3} (45^{\frac{1}{3} \cdot 3})$

Step 1.9.2

Cancel the common factor of $3$ .

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Step 1.9.2.1

Cancel the common factor.

$\log_{3} (36) - \log_{3} (20) + \log_{3} (45^{\frac{1}{3} \cdot 3})$

Step 1.9.2.2

Rewrite the expression.

$\log_{3} (36) - \log_{3} (20) + \log_{3} (45^{1})$

Step 1.10

Evaluate the exponent.

$\log_{3} (36) - \log_{3} (20) + \log_{3} (45)$

Step 2

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

$\log_{3} (\frac{36}{20}) + \log_{3} (45)$

Step 3

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

$\log_{3} (\frac{36}{20} \cdot 45)$

Step 4

Cancel the common factor of $5$ .

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Step 4.1

Factor $5$ out of $20$ .

$\log_{3} (\frac{36}{5 (4)} \cdot 45)$

Step 4.2

Factor $5$ out of $45$ .

$\log_{3} (\frac{36}{5 \cdot 4} \cdot (5 \cdot 9))$

Step 4.3

Cancel the common factor.

$\log_{3} (\frac{36}{5 \cdot 4} \cdot (5 \cdot 9))$

Step 4.4

Rewrite the expression.

$\log_{3} (\frac{36}{4} \cdot 9)$

Step 5

Combine $\frac{36}{4}$ and $9$ .

$\log_{3} (\frac{36 \cdot 9}{4})$

Step 6

Simplify the expression.

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Step 6.1

Multiply $36$ by $9$ .

$\log_{3} (\frac{324}{4})$

Step 6.2

Logarithm base $3$ of $\frac{324}{4}$ is $4$ .

$4$