Precalculus Examples

$\frac{1}{2} \cdot \log_{3} (144) - 2 \log_{3} (6)$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{2} \log_{3} (144)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{3} (144^{\frac{1}{2}}) - 2 \log_{3} (6)$

Step 1.2

Rewrite $144$ as $12^{2}$ .

$\log_{3} ({(12^{2})}^{\frac{1}{2}}) - 2 \log_{3} (6)$

Step 1.3

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

$\log_{3} (12^{2 (\frac{1}{2})}) - 2 \log_{3} (6)$

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Cancel the common factor.

$\log_{3} (12^{2 (\frac{1}{2})}) - 2 \log_{3} (6)$

Step 1.4.2

Rewrite the expression.

$\log_{3} (12^{1}) - 2 \log_{3} (6)$

Step 1.5

Evaluate the exponent.

$\log_{3} (12) - 2 \log_{3} (6)$

Step 1.6

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

$\log_{3} (12) - \log_{3} (6^{2})$

Step 1.7

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

$\log_{3} (12) - \log_{3} (36)$

Step 2

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

$\log_{3} (\frac{12}{36})$

Step 3

Reduce the expression $\frac{12}{36}$ by cancelling the common factors.

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Step 3.1

Factor $12$ out of $12$ .

$\log_{3} (\frac{12 (1)}{36})$

Step 3.2

Factor $12$ out of $36$ .

$\log_{3} (\frac{12 \cdot 1}{12 \cdot 3})$

Step 3.3

Cancel the common factor.

$\log_{3} (\frac{12 \cdot 1}{12 \cdot 3})$

Step 3.4

Rewrite the expression.

$\log_{3} (\frac{1}{3})$

Step 4

Rewrite $\frac{1}{3}$ as $1 \cdot 3^{- 1}$ .

$\log_{3} (1 \cdot 3^{- 1})$

Step 5

Rewrite $\log_{3} (1 \cdot 3^{- 1})$ as $\log_{3} (1) + \log_{3} (3^{- 1})$ .

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

Step 6

Use logarithm rules to move $- 1$ out of the exponent.

$\log_{3} (1) - \log_{3} (3)$

Step 7

Logarithm base $3$ of $3$ is $1$ .

$\log_{3} (1) - 1 \cdot 1$

Step 8

Multiply $- 1$ by $1$ .

$\log_{3} (1) - 1$

Step 9

Logarithm base $3$ of $1$ is $0$ .

$0 - 1$

Step 10

Subtract $1$ from $0$ .

$- 1$