Precalculus Examples

$\log_{3} (15) - \log_{3} (40) + \log_{3} (72)$

Step 1

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

$\log_{3} (\frac{15}{40}) + \log_{3} (72)$

Step 2

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

$\log_{3} (\frac{15}{40} \cdot 72)$

Step 3

Cancel the common factor of $8$ .

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Step 3.1

Factor $8$ out of $40$ .

$\log_{3} (\frac{15}{8 (5)} \cdot 72)$

Step 3.2

Factor $8$ out of $72$ .

$\log_{3} (\frac{15}{8 \cdot 5} \cdot (8 \cdot 9))$

Step 3.3

Cancel the common factor.

$\log_{3} (\frac{15}{8 \cdot 5} \cdot (8 \cdot 9))$

Step 3.4

Rewrite the expression.

$\log_{3} (\frac{15}{5} \cdot 9)$

Step 4

Combine $\frac{15}{5}$ and $9$ .

$\log_{3} (\frac{15 \cdot 9}{5})$

Step 5

Simplify the expression.

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Step 5.1

Multiply $15$ by $9$ .

$\log_{3} (\frac{135}{5})$

Step 5.2

Logarithm base $3$ of $\frac{135}{5}$ is $3$ .

$3$