Precalculus Examples

$\log_{5} (\frac{5}{36})$

Step 1

Rewrite $\log_{5} (\frac{5}{36})$ as $\log_{5} (5) - \log_{5} (36)$ .

$\log_{5} (5) - \log_{5} (36)$

Step 2

Logarithm base $5$ of $5$ is $1$ .

$1 - \log_{5} (36)$

Step 3

Rewrite $\log_{5} (36)$ as $\log_{5} (2^{2} \cdot 3^{2})$ .

$1 - \log_{5} (2^{2} \cdot 3^{2})$

Step 4

Rewrite $\log_{5} (2^{2} \cdot 3^{2})$ as $\log_{5} (2^{2}) + \log_{5} (3^{2})$ .

$1 - (\log_{5} (2^{2}) + \log_{5} (3^{2}))$

Step 5

Simplify each term.

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Step 5.1

Expand $\log_{5} (2^{2})$ by moving $2$ outside the logarithm.

$1 - (2 \log_{5} (2) + \log_{5} (3^{2}))$

Step 5.2

Expand $\log_{5} (3^{2})$ by moving $2$ outside the logarithm.

$1 - (2 \log_{5} (2) + 2 \log_{5} (3))$

Step 6

Apply the distributive property.

$1 - (2 \log_{5} (2)) - (2 \log_{5} (3))$

Step 7

Multiply $2$ by $- 1$ .

$1 - 2 \log_{5} (2) - (2 \log_{5} (3))$

Step 8

Multiply $2$ by $- 1$ .

$1 - 2 \log_{5} (2) - 2 \log_{5} (3)$