Precalculus Examples

$\log_{5} (\frac{25}{125})$

Step 1

Reduce the expression $\frac{25}{125}$ by cancelling the common factors.

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Step 1.1

Factor $25$ out of $25$ .

$\log_{5} (\frac{25 (1)}{125})$

Step 1.2

Factor $25$ out of $125$ .

$\log_{5} (\frac{25 \cdot 1}{25 \cdot 5})$

Step 1.3

Cancel the common factor.

$\log_{5} (\frac{25 \cdot 1}{25 \cdot 5})$

Step 1.4

Rewrite the expression.

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

Step 2

Rewrite $\frac{1}{5}$ as $1 \cdot 5^{- 1}$ .

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

Step 3

Rewrite $\log_{5} (1 \cdot 5^{- 1})$ as $\log_{5} (1) + \log_{5} (5^{- 1})$ .

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

Step 4

Use logarithm rules to move $- 1$ out of the exponent.

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

Step 5

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

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

Step 6

Multiply $- 1$ by $1$ .

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

Step 7

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

$0 - 1$

Step 8

Subtract $1$ from $0$ .

$- 1$