Finite Math Examples

$\log_{3} (\sqrt{\frac{x}{a^{4}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x}{a^{4}}}$ as ${(\frac{x}{a^{4}})}^{\frac{1}{2}}$ .

$\log_{3} ({(\frac{x}{a^{4}})}^{\frac{1}{2}})$

Step 2

Expand $\log_{3} ({(\frac{x}{a^{4}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{3} (\frac{x}{a^{4}})$

Step 3

Rewrite $\log_{3} (\frac{x}{a^{4}})$ as $\log_{3} (x) - \log_{3} (a^{4})$ .

$\frac{1}{2} (\log_{3} (x) - \log_{3} (a^{4}))$

Step 4

Expand $\log_{3} (a^{4})$ by moving $4$ outside the logarithm.

$\frac{1}{2} (\log_{3} (x) - (4 \log_{3} (a)))$

Step 5

Multiply $4$ by $- 1$ .

$\frac{1}{2} (\log_{3} (x) - 4 \log_{3} (a))$

Step 6

Apply the distributive property.

$\frac{1}{2} \log_{3} (x) + \frac{1}{2} (- 4 \log_{3} (a))$

Step 7

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

$\frac{\log_{3} (x)}{2} + \frac{1}{2} (- 4 \log_{3} (a))$

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Factor $2$ out of $- 4 \log_{3} (a)$ .

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

$\frac{\log_{3} (x)}{2} - 2 \log_{3} (a)$