Precalculus Examples

$\log_{3} (\frac{6 \sqrt{x}}{y^{5}})$

Step 1

Rewrite $\log_{3} (\frac{6 \sqrt{x}}{y^{5}})$ as $\log_{3} (6 \sqrt{x}) - \log_{3} (y^{5})$ .

$\log_{3} (6 \sqrt{x}) - \log_{3} (y^{5})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\log_{3} (6 x^{\frac{1}{2}}) - \log_{3} (y^{5})$

Step 3

Expand $\log_{3} (y^{5})$ by moving $5$ outside the logarithm.

$\log_{3} (6 x^{\frac{1}{2}}) - (5 \log_{3} (y))$

Step 4

Multiply $5$ by $- 1$ .

$\log_{3} (6 x^{\frac{1}{2}}) - 5 \log_{3} (y)$

Step 5

Rewrite $\log_{3} (6 x^{\frac{1}{2}})$ as $\log_{3} (6) + \log_{3} (x^{\frac{1}{2}})$ .

$\log_{3} (6) + \log_{3} (x^{\frac{1}{2}}) - 5 \log_{3} (y)$

Step 6

Expand $\log_{3} (x^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{3} (6) + \frac{1}{2} \log_{3} (x) - 5 \log_{3} (y)$

Step 7

Simplify each term.

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Step 7.1

Rewrite $\log_{3} (6)$ as $\log_{3} (2 (3))$ .

$\log_{3} (2 (3)) + \frac{1}{2} \log_{3} (x) - 5 \log_{3} (y)$

Step 7.2

Rewrite $\log_{3} (2 (3))$ as $\log_{3} (2) + \log_{3} (3)$ .

$\log_{3} (2) + \log_{3} (3) + \frac{1}{2} \log_{3} (x) - 5 \log_{3} (y)$

Step 7.3

Logarithm base $3$ of $3$ is $1$ .

$\log_{3} (2) + 1 + \frac{1}{2} \log_{3} (x) - 5 \log_{3} (y)$

Step 7.4

Combine $\frac{1}{2}$ and $\log_{3} (x)$ .

$\log_{3} (2) + 1 + \frac{\log_{3} (x)}{2} - 5 \log_{3} (y)$