Precalculus Examples

$\log_{3} (\sqrt{9 x y})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 x y}$ as ${(9 x y)}^{\frac{1}{2}}$ .

$\log_{3} ({(9 x y)}^{\frac{1}{2}})$

Step 2

Expand $\log_{3} ({(9 x y)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{3} (9 x y)$

Step 3

Rewrite $\log_{3} (9 x y)$ as $\log_{3} (9 x) + \log_{3} (y)$ .

$\frac{1}{2} (\log_{3} (9 x) + \log_{3} (y))$

Step 4

Rewrite $\log_{3} (9 x)$ as $\log_{3} (9) + \log_{3} (x)$ .

$\frac{1}{2} (\log_{3} (9) + \log_{3} (x) + \log_{3} (y))$

Step 5

Logarithm base $3$ of $9$ is $2$ .

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

Step 6

Apply the distributive property.

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

Step 7

Simplify.

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Step 7.1

Cancel the common factor of $2$ .

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Step 7.1.1

Cancel the common factor.

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

Step 7.1.2

Rewrite the expression.

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

Step 7.2

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

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

Step 7.3

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

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