Precalculus Examples

$\log (\frac{1}{36}) + \log (64)$

Step 1

Rewrite $\log (\frac{1}{36})$ as $\log (1) - \log (36)$ .

$\log (1) - \log (36) + \log (64)$

Step 2

Logarithm base $10$ of $1$ is $0$ .

$0 - \log (36) + \log (64)$

Step 3

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

$0 - \log (2^{2} \cdot 3^{2}) + \log (64)$

Step 4

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

$0 - (\log (2^{2}) + \log (3^{2})) + \log (64)$

Step 5

Simplify each term.

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Step 5.1

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

$0 - (2 \log (2) + \log (3^{2})) + \log (64)$

Step 5.2

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

$0 - (2 \log (2) + 2 \log (3)) + \log (64)$

Step 6

Apply the distributive property.

$0 - (2 \log (2)) - (2 \log (3)) + \log (64)$

Step 7

Multiply $2$ by $- 1$ .

$0 - 2 \log (2) - (2 \log (3)) + \log (64)$

Step 8

Multiply $2$ by $- 1$ .

$0 - 2 \log (2) - 2 \log (3) + \log (64)$

Step 9

Subtract $2 \log (2)$ from $0$ .

$- 2 \log (2) - 2 \log (3) + \log (64)$

Step 10

Simplify each term.

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Step 10.1

Rewrite $\log (64)$ as $\log (2^{6})$ .

$- 2 \log (2) - 2 \log (3) + \log (2^{6})$

Step 10.2

Expand $\log (2^{6})$ by moving $6$ outside the logarithm.

$- 2 \log (2) - 2 \log (3) + 6 \log (2)$

Step 11

Add $- 2 \log (2)$ and $6 \log (2)$ .

$4 \log (2) - 2 \log (3)$