Precalculus Examples

$\log_{6} (\sqrt[4]{4 x^{3}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{4 x^{3}}$ as ${(4 x^{3})}^{\frac{1}{4}}$ .

$\log_{6} ({(4 x^{3})}^{\frac{1}{4}})$

Step 2

Expand $\log_{6} ({(4 x^{3})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \log_{6} (4 x^{3})$

Step 3

Rewrite $\log_{6} (4 x^{3})$ as $\log_{6} (4) + \log_{6} (x^{3})$ .

$\frac{1}{4} (\log_{6} (4) + \log_{6} (x^{3}))$

Step 4

Expand $\log_{6} (x^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{4} (\log_{6} (4) + 3 \log_{6} (x))$

Step 5

Simplify each term.

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Step 5.1

Rewrite $\log_{6} (4)$ as $\log_{6} (2^{2})$ .

$\frac{1}{4} (\log_{6} (2^{2}) + 3 \log_{6} (x))$

Step 5.2

Expand $\log_{6} (2^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{4} (2 \log_{6} (2) + 3 \log_{6} (x))$

Step 6

Apply the distributive property.

$\frac{1}{4} (2 \log_{6} (2)) + \frac{1}{4} (3 \log_{6} (x))$

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Factor $2$ out of $4$ .

$\frac{1}{2 (2)} (2 \log_{6} (2)) + \frac{1}{4} (3 \log_{6} (x))$

Step 7.2

Factor $2$ out of $2 \log_{6} (2)$ .

$\frac{1}{2 (2)} (2 (\log_{6} (2))) + \frac{1}{4} (3 \log_{6} (x))$

Step 7.3

Cancel the common factor.

$\frac{1}{2 \cdot 2} (2 \log_{6} (2)) + \frac{1}{4} (3 \log_{6} (x))$

Step 7.4

Rewrite the expression.

$\frac{1}{2} \log_{6} (2) + \frac{1}{4} (3 \log_{6} (x))$

Step 8

Combine $\frac{1}{2}$ and $\log_{6} (2)$ .

$\frac{\log_{6} (2)}{2} + \frac{1}{4} (3 \log_{6} (x))$

Step 9

Multiply $\frac{1}{4} (3 \log_{6} (x))$ .

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Step 9.1

Combine $3$ and $\frac{1}{4}$ .

$\frac{\log_{6} (2)}{2} + \frac{3}{4} \log_{6} (x)$

Step 9.2

Combine $\frac{3}{4}$ and $\log_{6} (x)$ .

$\frac{\log_{6} (2)}{2} + \frac{3 \log_{6} (x)}{4}$