Precalculus Examples

$\log_{3} (\frac{\sqrt{27 x^{5}}}{\sqrt[4]{9 y^{3}}})$

Step 1

Rewrite $\log_{3} (\frac{\sqrt{27 x^{5}}}{\sqrt[4]{9 y^{3}}})$ as $\log_{3} (\sqrt{27 x^{5}}) - \log_{3} (\sqrt[4]{9 y^{3}})$ .

$\log_{3} (\sqrt{27 x^{5}}) - \log_{3} (\sqrt[4]{9 y^{3}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{27 x^{5}}$ as ${(27 x^{5})}^{\frac{1}{2}}$ .

$\log_{3} ({(27 x^{5})}^{\frac{1}{2}}) - \log_{3} (\sqrt[4]{9 y^{3}})$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{9 y^{3}}$ as ${(9 y^{3})}^{\frac{1}{4}}$ .

$\log_{3} ({(27 x^{5})}^{\frac{1}{2}}) - \log_{3} ({(9 y^{3})}^{\frac{1}{4}})$

Step 4

Expand $\log_{3} ({(27 x^{5})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{3} (27 x^{5}) - \log_{3} ({(9 y^{3})}^{\frac{1}{4}})$

Step 5

Expand $\log_{3} ({(9 y^{3})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{2} \log_{3} (27 x^{5}) - (\frac{1}{4} \log_{3} (9 y^{3}))$

Step 6

Combine $\frac{1}{4}$ and $\log_{3} (9 y^{3})$ .

$\frac{1}{2} \log_{3} (27 x^{5}) - \frac{\log_{3} (9 y^{3})}{4}$

Step 7

Rewrite $\log_{3} (27 x^{5})$ as $\log_{3} (27) + \log_{3} (x^{5})$ .

$\frac{1}{2} (\log_{3} (27) + \log_{3} (x^{5})) - \frac{\log_{3} (9 y^{3})}{4}$

Step 8

Expand $\log_{3} (x^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{2} (\log_{3} (27) + 5 \log_{3} (x)) - \frac{\log_{3} (9 y^{3})}{4}$

Step 9

Logarithm base $3$ of $27$ is $3$ .

$\frac{1}{2} (3 + 5 \log_{3} (x)) - \frac{\log_{3} (9 y^{3})}{4}$

Step 10

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

$\frac{1}{2} (3 + 5 \log_{3} (x)) - \frac{\log_{3} (9) + \log_{3} (y^{3})}{4}$

Step 11

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

$\frac{1}{2} (3 + 5 \log_{3} (x)) - \frac{\log_{3} (9) + 3 \log_{3} (y)}{4}$

Step 12

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

$\frac{1}{2} (3 + 5 \log_{3} (x)) - \frac{2 + 3 \log_{3} (y)}{4}$

Step 13

Simplify each term.

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Step 13.1

Apply the distributive property.

$\frac{1}{2} \cdot 3 + \frac{1}{2} (5 \log_{3} (x)) - \frac{2 + 3 \log_{3} (y)}{4}$

Step 13.2

Combine $\frac{1}{2}$ and $3$ .

$\frac{3}{2} + \frac{1}{2} (5 \log_{3} (x)) - \frac{2 + 3 \log_{3} (y)}{4}$

Step 13.3

Multiply $\frac{1}{2} (5 \log_{3} (x))$ .

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Step 13.3.1

Combine $5$ and $\frac{1}{2}$ .

$\frac{3}{2} + \frac{5}{2} \log_{3} (x) - \frac{2 + 3 \log_{3} (y)}{4}$

Step 13.3.2

Combine $\frac{5}{2}$ and $\log_{3} (x)$ .

$\frac{3}{2} + \frac{5 \log_{3} (x)}{2} - \frac{2 + 3 \log_{3} (y)}{4}$

Step 14

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{2} \cdot \frac{2}{2} - \frac{2 + 3 \log_{3} (y)}{4} + \frac{5 \log_{3} (x)}{2}$

Step 15

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 15.1

Multiply $\frac{3}{2}$ by $\frac{2}{2}$ .

$\frac{3 \cdot 2}{2 \cdot 2} - \frac{2 + 3 \log_{3} (y)}{4} + \frac{5 \log_{3} (x)}{2}$

Step 15.2

Multiply $2$ by $2$ .

$\frac{3 \cdot 2}{4} - \frac{2 + 3 \log_{3} (y)}{4} + \frac{5 \log_{3} (x)}{2}$

Step 16

Combine the numerators over the common denominator.

$\frac{3 \cdot 2 - (2 + 3 \log_{3} (y))}{4} + \frac{5 \log_{3} (x)}{2}$

Step 17

Multiply $3$ by $2$ .

$\frac{6 - (2 + 3 \log_{3} (y))}{4} + \frac{5 \log_{3} (x)}{2}$

Step 18

Simplify the numerator.

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Step 18.1

Apply the distributive property.

$\frac{6 - 1 \cdot 2 - (3 \log_{3} (y))}{4} + \frac{5 \log_{3} (x)}{2}$

Step 18.2

Multiply $- 1$ by $2$ .

$\frac{6 - 2 - (3 \log_{3} (y))}{4} + \frac{5 \log_{3} (x)}{2}$

Step 18.3

Multiply $3$ by $- 1$ .

$\frac{6 - 2 - 3 \log_{3} (y)}{4} + \frac{5 \log_{3} (x)}{2}$

Step 18.4

Subtract $2$ from $6$ .

$\frac{4 - 3 \log_{3} (y)}{4} + \frac{5 \log_{3} (x)}{2}$

Step 19

To write $\frac{5 \log_{3} (x)}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{4 - 3 \log_{3} (y)}{4} + \frac{5 \log_{3} (x)}{2} \cdot \frac{2}{2}$

Step 20

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 20.1

Multiply $\frac{5 \log_{3} (x)}{2}$ by $\frac{2}{2}$ .

$\frac{4 - 3 \log_{3} (y)}{4} + \frac{5 \log_{3} (x) \cdot 2}{2 \cdot 2}$

Step 20.2

Multiply $2$ by $2$ .

$\frac{4 - 3 \log_{3} (y)}{4} + \frac{5 \log_{3} (x) \cdot 2}{4}$

Step 21

Combine the numerators over the common denominator.

$\frac{4 - 3 \log_{3} (y) + 5 \log_{3} (x) \cdot 2}{4}$

Step 22

Multiply $2$ by $5$ .

$\frac{4 - 3 \log_{3} (y) + 10 \log_{3} (x)}{4}$