Precalculus Examples

$\log_{3} (\frac{x^{\frac{2}{7}} y^{\frac{19}{2}}}{27})$

Step 1

Rewrite $\log_{3} (\frac{x^{\frac{2}{7}} y^{\frac{19}{2}}}{27})$ as $\log_{3} (x^{\frac{2}{7}} y^{\frac{19}{2}}) - \log_{3} (27)$ .

$\log_{3} (x^{\frac{2}{7}} y^{\frac{19}{2}}) - \log_{3} (27)$

Step 2

Logarithm base $3$ of $27$ is $3$ .

$\log_{3} (x^{\frac{2}{7}} y^{\frac{19}{2}}) - 1 \cdot 3$

Step 3

Multiply $- 1$ by $3$ .

$\log_{3} (x^{\frac{2}{7}} y^{\frac{19}{2}}) - 3$

Step 4

Rewrite $\log_{3} (x^{\frac{2}{7}} y^{\frac{19}{2}})$ as $\log_{3} (x^{\frac{2}{7}}) + \log_{3} (y^{\frac{19}{2}})$ .

$\log_{3} (x^{\frac{2}{7}}) + \log_{3} (y^{\frac{19}{2}}) - 3$

Step 5

Expand $\log_{3} (x^{\frac{2}{7}})$ by moving $\frac{2}{7}$ outside the logarithm.

$\frac{2}{7} \log_{3} (x) + \log_{3} (y^{\frac{19}{2}}) - 3$

Step 6

Expand $\log_{3} (y^{\frac{19}{2}})$ by moving $\frac{19}{2}$ outside the logarithm.

$\frac{2}{7} \log_{3} (x) + \frac{19}{2} \log_{3} (y) - 3$

Step 7

Simplify each term.

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Step 7.1

Combine $\frac{2}{7}$ and $\log_{3} (x)$ .

$\frac{2 \log_{3} (x)}{7} + \frac{19}{2} \log_{3} (y) - 3$

Step 7.2

Combine $\frac{19}{2}$ and $\log_{3} (y)$ .

$\frac{2 \log_{3} (x)}{7} + \frac{19 \log_{3} (y)}{2} - 3$