Precalculus Examples

$\log_{5} (\frac{1}{50})$

Step 1

Rewrite $\log_{5} (\frac{1}{50})$ as $\log_{5} (1) - \log_{5} (50)$ .

$\log_{5} (1) - \log_{5} (50)$

Step 2

Logarithm base $5$ of $1$ is $0$ .

$0 - \log_{5} (50)$

Step 3

Rewrite $\log_{5} (50)$ as $\log_{5} (2 \cdot 5^{2})$ .

$0 - \log_{5} (2 \cdot 5^{2})$

Step 4

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

$0 - (\log_{5} (2) + \log_{5} (5^{2}))$

Step 5

Use logarithm rules to move $2$ out of the exponent.

$0 - (\log_{5} (2) + 2 \log_{5} (5))$

Step 6

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

$0 - (\log_{5} (2) + 2 \cdot 1)$

Step 7

Multiply $2$ by $1$ .

$0 - (\log_{5} (2) + 2)$

Step 8

Apply the distributive property.

$0 - \log_{5} (2) - 1 \cdot 2$

Step 9

Multiply $- 1$ by $2$ .

$0 - \log_{5} (2) - 2$

Step 10

Subtract $\log_{5} (2)$ from $0$ .

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