Precalculus Examples

$\log_{2} (\frac{\sqrt{4 x^{6}}}{\sqrt[6]{32 y^{3}}})$

Step 1

Rewrite $\log_{2} (\frac{\sqrt{4 x^{6}}}{\sqrt[6]{32 y^{3}}})$ as $\log_{2} (\sqrt{4 x^{6}}) - \log_{2} (\sqrt[6]{32 y^{3}})$ .

$\log_{2} (\sqrt{4 x^{6}}) - \log_{2} (\sqrt[6]{32 y^{3}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x^{6}}$ as ${(4 x^{6})}^{\frac{1}{2}}$ .

$\log_{2} ({(4 x^{6})}^{\frac{1}{2}}) - \log_{2} (\sqrt[6]{32 y^{3}})$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{32 y^{3}}$ as ${(32 y^{3})}^{\frac{1}{6}}$ .

$\log_{2} ({(4 x^{6})}^{\frac{1}{2}}) - \log_{2} ({(32 y^{3})}^{\frac{1}{6}})$

Step 4

Expand $\log_{2} ({(4 x^{6})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{2} (4 x^{6}) - \log_{2} ({(32 y^{3})}^{\frac{1}{6}})$

Step 5

Expand $\log_{2} ({(32 y^{3})}^{\frac{1}{6}})$ by moving $\frac{1}{6}$ outside the logarithm.

$\frac{1}{2} \log_{2} (4 x^{6}) - (\frac{1}{6} \log_{2} (32 y^{3}))$

Step 6

Combine $\frac{1}{6}$ and $\log_{2} (32 y^{3})$ .

$\frac{1}{2} \log_{2} (4 x^{6}) - \frac{\log_{2} (32 y^{3})}{6}$

Step 7

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

$\frac{1}{2} (\log_{2} (4) + \log_{2} (x^{6})) - \frac{\log_{2} (32 y^{3})}{6}$

Step 8

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

$\frac{1}{2} (\log_{2} (4) + 6 \log_{2} (x)) - \frac{\log_{2} (32 y^{3})}{6}$

Step 9

Logarithm base $2$ of $4$ is $2$ .

$\frac{1}{2} (2 + 6 \log_{2} (x)) - \frac{\log_{2} (32 y^{3})}{6}$

Step 10

Rewrite $\log_{2} (32 y^{3})$ as $\log_{2} (32) + \log_{2} (y^{3})$ .

$\frac{1}{2} (2 + 6 \log_{2} (x)) - \frac{\log_{2} (32) + \log_{2} (y^{3})}{6}$

Step 11

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

$\frac{1}{2} (2 + 6 \log_{2} (x)) - \frac{\log_{2} (32) + 3 \log_{2} (y)}{6}$

Step 12

Logarithm base $2$ of $32$ is $5$ .

$\frac{1}{2} (2 + 6 \log_{2} (x)) - \frac{5 + 3 \log_{2} (y)}{6}$

Step 13

Simplify each term.

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Step 13.1

Apply the distributive property.

$\frac{1}{2} \cdot 2 + \frac{1}{2} (6 \log_{2} (x)) - \frac{5 + 3 \log_{2} (y)}{6}$

Step 13.2

Cancel the common factor of $2$ .

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Step 13.2.1

Cancel the common factor.

$\frac{1}{2} \cdot 2 + \frac{1}{2} (6 \log_{2} (x)) - \frac{5 + 3 \log_{2} (y)}{6}$

Step 13.2.2

Rewrite the expression.

$1 + \frac{1}{2} (6 \log_{2} (x)) - \frac{5 + 3 \log_{2} (y)}{6}$

Step 13.3

Cancel the common factor of $2$ .

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Step 13.3.1

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

$1 + \frac{1}{2} (2 (3 \log_{2} (x))) - \frac{5 + 3 \log_{2} (y)}{6}$

Step 13.3.2

Cancel the common factor.

$1 + \frac{1}{2} (2 (3 \log_{2} (x))) - \frac{5 + 3 \log_{2} (y)}{6}$

Step 13.3.3

Rewrite the expression.

$1 + 3 \log_{2} (x) - \frac{5 + 3 \log_{2} (y)}{6}$

Step 14

Write $1$ as a fraction with a common denominator.

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

Step 15

Combine the numerators over the common denominator.

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

Step 16

Simplify the numerator.

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Step 16.1

Apply the distributive property.

$3 \log_{2} (x) + \frac{6 - 1 \cdot 5 - (3 \log_{2} (y))}{6}$

Step 16.2

Multiply $- 1$ by $5$ .

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

Step 16.3

Multiply $3$ by $- 1$ .

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

Step 16.4

Subtract $5$ from $6$ .

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

Step 17

To write $3 \log_{2} (x)$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Step 18

Combine $3 \log_{2} (x)$ and $\frac{6}{6}$ .

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

Step 19

Combine the numerators over the common denominator.

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

Step 20

Multiply $6$ by $3$ .

$\frac{18 \log_{2} (x) + 1 - 3 \log_{2} (y)}{6}$