Precalculus Examples

$\log_{4} (36) - \log_{4} (9)$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\log_{4} (36)$ as $\log_{4} (2^{2} \cdot 3^{2})$ .

$\log_{4} (2^{2} \cdot 3^{2}) - \log_{4} (9)$

Step 1.2

Rewrite $\log_{4} (2^{2} \cdot 3^{2})$ as $\log_{4} (2^{2}) + \log_{4} (3^{2})$ .

$\log_{4} (2^{2}) + \log_{4} (3^{2}) - \log_{4} (9)$

Step 1.3

Simplify each term.

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Step 1.3.1

Expand $\log_{4} (2^{2})$ by moving $2$ outside the logarithm.

$2 \log_{4} (2) + \log_{4} (3^{2}) - \log_{4} (9)$

Step 1.3.2

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

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

Step 1.3.3

Cancel the common factor of $2$ .

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Step 1.3.3.1

Cancel the common factor.

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

Step 1.3.3.2

Rewrite the expression.

$1 + \log_{4} (3^{2}) - \log_{4} (9)$

Step 1.3.4

Expand $\log_{4} (3^{2})$ by moving $2$ outside the logarithm.

$1 + 2 \log_{4} (3) - \log_{4} (9)$

Step 1.4

Rewrite $\log_{4} (9)$ as $\log_{4} (3^{2})$ .

$1 + 2 \log_{4} (3) - \log_{4} (3^{2})$

Step 1.5

Expand $\log_{4} (3^{2})$ by moving $2$ outside the logarithm.

$1 + 2 \log_{4} (3) - (2 \log_{4} (3))$

Step 1.6

Multiply $2$ by $- 1$ .

$1 + 2 \log_{4} (3) - 2 \log_{4} (3)$

Step 2

Subtract $2 \log_{4} (3)$ from $2 \log_{4} (3)$ .

$1 + 0$

Step 3

Add $1$ and $0$ .

$1$