Precalculus Examples

$\log_{3} ({(27 x^{5} \sqrt[3]{y})}^{2})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y}$ as $y^{\frac{1}{3}}$ .

$\log_{3} ({(27 x^{5} y^{\frac{1}{3}})}^{2})$

Step 2

Expand $\log_{3} ({(27 x^{5} y^{\frac{1}{3}})}^{2})$ by moving $2$ outside the logarithm.

$2 \log_{3} (27 x^{5} y^{\frac{1}{3}})$

Step 3

Rewrite $\log_{3} (27 x^{5} y^{\frac{1}{3}})$ as $\log_{3} (27 x^{5}) + \log_{3} (y^{\frac{1}{3}})$ .

$2 (\log_{3} (27 x^{5}) + \log_{3} (y^{\frac{1}{3}}))$

Step 4

Rewrite $\log_{3} (27 x^{5})$ as $\log_{3} (27) + \log_{3} (x^{5})$ .

$2 (\log_{3} (27) + \log_{3} (x^{5}) + \log_{3} (y^{\frac{1}{3}}))$

Step 5

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

$2 (\log_{3} (27) + 5 \log_{3} (x) + \log_{3} (y^{\frac{1}{3}}))$

Step 6

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

$2 (\log_{3} (27) + 5 \log_{3} (x) + \frac{1}{3} \log_{3} (y))$

Step 7

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

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

Step 8

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

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

Step 9

Apply the distributive property.

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

Step 10

Simplify.

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Step 10.1

Multiply $2$ by $3$ .

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

Step 10.2

Multiply $5$ by $2$ .

$6 + 10 \log_{3} (x) + 2 \frac{\log_{3} (y)}{3}$

Step 10.3

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

$6 + 10 \log_{3} (x) + \frac{2 \log_{3} (y)}{3}$