Precalculus Examples

$\log_{5} (3) + \log_{5} (2) - \log_{5} (3750)$

Step 1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{5} (3 \cdot 2) - \log_{5} (3750)$

Step 2

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

$\log_{5} (\frac{3 \cdot 2}{3750})$

Step 3

Cancel the common factor of $3$ and $3750$ .

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Step 3.1

Factor $3$ out of $3 \cdot 2$ .

$\log_{5} (\frac{3 (2)}{3750})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $3$ out of $3750$ .

$\log_{5} (\frac{3 \cdot 2}{3 \cdot 1250})$

Step 3.2.2

Cancel the common factor.

$\log_{5} (\frac{3 \cdot 2}{3 \cdot 1250})$

Step 3.2.3

Rewrite the expression.

$\log_{5} (\frac{2}{1250})$

Step 4

Cancel the common factor of $2$ and $1250$ .

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Step 4.1

Factor $2$ out of $2$ .

$\log_{5} (\frac{2 (1)}{1250})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $2$ out of $1250$ .

$\log_{5} (\frac{2 \cdot 1}{2 \cdot 625})$

Step 4.2.2

Cancel the common factor.

$\log_{5} (\frac{2 \cdot 1}{2 \cdot 625})$

Step 4.2.3

Rewrite the expression.

$\log_{5} (\frac{1}{625})$

Step 5

Logarithm base $5$ of $\frac{1}{625}$ is $- 4$ .

$- 4$