Calculus Examples

$\log_{5} (180) - \log_{5} (75) - \log_{5} (60)$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{5} (\frac{180}{75}) - \log_{5} (60)$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{5} (\frac{\frac{180}{75}}{60})$

Step 3

Cancel the common factor of $180$ and $75$ .

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Step 3.1

Factor $15$ out of $180$ .

$\log_{5} (\frac{\frac{15 (12)}{75}}{60})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $15$ out of $75$ .

$\log_{5} (\frac{\frac{15 \cdot 12}{15 \cdot 5}}{60})$

Step 3.2.2

Cancel the common factor.

$\log_{5} (\frac{\frac{15 \cdot 12}{15 \cdot 5}}{60})$

Step 3.2.3

Rewrite the expression.

$\log_{5} (\frac{\frac{12}{5}}{60})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log_{5} (\frac{12}{5} \cdot \frac{1}{60})$

Step 5

Cancel the common factor of $12$ .

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Step 5.1

Factor $12$ out of $60$ .

$\log_{5} (\frac{12}{5} \cdot \frac{1}{12 (5)})$

Step 5.2

Cancel the common factor.

$\log_{5} (\frac{12}{5} \cdot \frac{1}{12 \cdot 5})$

Step 5.3

Rewrite the expression.

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

Step 6

Multiply $\frac{1}{5}$ by $\frac{1}{5}$ .

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

Step 7

Simplify the expression.

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Step 7.1

Multiply $5$ by $5$ .

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

Step 7.2

Logarithm base $5$ of $\frac{1}{25}$ is $- 2$ .

$- 2$