Precalculus Examples

$\log_{4} (\sqrt{10 x})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 x}$ as ${(10 x)}^{\frac{1}{2}}$ .

$\log_{4} ({(10 x)}^{\frac{1}{2}})$

Step 2

Expand $\log_{4} ({(10 x)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{4} (10 x)$

Step 3

Rewrite $\log_{4} (10 x)$ as $\log_{4} (10) + \log_{4} (x)$ .

$\frac{1}{2} (\log_{4} (10) + \log_{4} (x))$

Step 4

Simplify each term.

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Step 4.1

Rewrite $\log_{4} (10)$ as $\log_{4} (2 (5))$ .

$\frac{1}{2} (\log_{4} (2 (5)) + \log_{4} (x))$

Step 4.2

Rewrite $\log_{4} (2 (5))$ as $\log_{4} (2) + \log_{4} (5)$ .

$\frac{1}{2} (\log_{4} (2) + \log_{4} (5) + \log_{4} (x))$

Step 4.3

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

$\frac{1}{2} (\frac{1}{2} + \log_{4} (5) + \log_{4} (x))$

Step 5

Apply the distributive property.

$\frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \log_{4} (5) + \frac{1}{2} \log_{4} (x)$

Step 6

Simplify.

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Step 6.1

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 6.1.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\frac{1}{2 \cdot 2} + \frac{1}{2} \log_{4} (5) + \frac{1}{2} \log_{4} (x)$

Step 6.1.2

Multiply $2$ by $2$ .

$\frac{1}{4} + \frac{1}{2} \log_{4} (5) + \frac{1}{2} \log_{4} (x)$

Step 6.2

Combine $\frac{1}{2}$ and $\log_{4} (5)$ .

$\frac{1}{4} + \frac{\log_{4} (5)}{2} + \frac{1}{2} \log_{4} (x)$

Step 6.3

Combine $\frac{1}{2}$ and $\log_{4} (x)$ .

$\frac{1}{4} + \frac{\log_{4} (5)}{2} + \frac{\log_{4} (x)}{2}$