Precalculus Examples

$\log_{3} (16) - 2 \log_{3} (36)$

Step 1

Simplify each term.

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Step 1.1

Simplify $- 2 \log_{3} (36)$ by moving $2$ inside the logarithm.

$\log_{3} (16) - \log_{3} (36^{2})$

Step 1.2

Raise $36$ to the power of $2$ .

$\log_{3} (16) - \log_{3} (1296)$

Step 2

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

$\log_{3} (\frac{16}{1296})$

Step 3

Cancel the common factor of $16$ and $1296$ .

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Step 3.1

Factor $16$ out of $16$ .

$\log_{3} (\frac{16 (1)}{1296})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $16$ out of $1296$ .

$\log_{3} (\frac{16 \cdot 1}{16 \cdot 81})$

Step 3.2.2

Cancel the common factor.

$\log_{3} (\frac{16 \cdot 1}{16 \cdot 81})$

Step 3.2.3

Rewrite the expression.

$\log_{3} (\frac{1}{81})$

Step 4

Logarithm base $3$ of $\frac{1}{81}$ is $- 4$ .

$- 4$