Calculus Examples

$\log_{a} (\sqrt[4]{\frac{m^{8} n^{12}}{a^{3} b^{5}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{\frac{m^{8} n^{12}}{a^{3} b^{5}}}$ as ${(\frac{m^{8} n^{12}}{a^{3} b^{5}})}^{\frac{1}{4}}$ .

$\log_{a} ({(\frac{m^{8} n^{12}}{a^{3} b^{5}})}^{\frac{1}{4}})$

Step 2

Expand $\log_{a} ({(\frac{m^{8} n^{12}}{a^{3} b^{5}})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \log_{a} (\frac{m^{8} n^{12}}{a^{3} b^{5}})$

Step 3

Rewrite $\log_{a} (\frac{m^{8} n^{12}}{a^{3} b^{5}})$ as $\log_{a} (m^{8} n^{12}) - \log_{a} (a^{3} b^{5})$ .

$\frac{1}{4} (\log_{a} (m^{8} n^{12}) - \log_{a} (a^{3} b^{5}))$

Step 4

Rewrite $\log_{a} (m^{8} n^{12})$ as $\log_{a} (m^{8}) + \log_{a} (n^{12})$ .

$\frac{1}{4} (\log_{a} (m^{8}) + \log_{a} (n^{12}) - \log_{a} (a^{3} b^{5}))$

Step 5

Expand $\log_{a} (m^{8})$ by moving $8$ outside the logarithm.

$\frac{1}{4} (8 \log_{a} (m) + \log_{a} (n^{12}) - \log_{a} (a^{3} b^{5}))$

Step 6

Expand $\log_{a} (n^{12})$ by moving $12$ outside the logarithm.

$\frac{1}{4} (8 \log_{a} (m) + 12 \log_{a} (n) - \log_{a} (a^{3} b^{5}))$

Step 7

Rewrite $\log_{a} (a^{3} b^{5})$ as $\log_{a} (a^{3}) + \log_{a} (b^{5})$ .

$\frac{1}{4} (8 \log_{a} (m) + 12 \log_{a} (n) - (\log_{a} (a^{3}) + \log_{a} (b^{5})))$

Step 8

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

$\frac{1}{4} (8 \log_{a} (m) + 12 \log_{a} (n) - (3 \log_{a} (a) + \log_{a} (b^{5})))$

Step 9

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

$\frac{1}{4} (8 \log_{a} (m) + 12 \log_{a} (n) - (3 \log_{a} (a) + 5 \log_{a} (b)))$

Step 10

Logarithm base $a$ of $a$ is $1$ .

$\frac{1}{4} (8 \log_{a} (m) + 12 \log_{a} (n) - (3 \cdot 1 + 5 \log_{a} (b)))$

Step 11

Multiply $3$ by $1$ .

$\frac{1}{4} (8 \log_{a} (m) + 12 \log_{a} (n) - (3 + 5 \log_{a} (b)))$

Step 12

Simplify each term.

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Step 12.1

Apply the distributive property.

$\frac{1}{4} (8 \log_{a} (m) + 12 \log_{a} (n) - 1 \cdot 3 - (5 \log_{a} (b)))$

Step 12.2

Multiply $- 1$ by $3$ .

$\frac{1}{4} (8 \log_{a} (m) + 12 \log_{a} (n) - 3 - (5 \log_{a} (b)))$

Step 12.3

Multiply $5$ by $- 1$ .

$\frac{1}{4} (8 \log_{a} (m) + 12 \log_{a} (n) - 3 - 5 \log_{a} (b))$

Step 13

Apply the distributive property.

$\frac{1}{4} (8 \log_{a} (m)) + \frac{1}{4} (12 \log_{a} (n)) + \frac{1}{4} \cdot - 3 + \frac{1}{4} (- 5 \log_{a} (b))$

Step 14

Simplify.

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Step 14.1

Cancel the common factor of $4$ .

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Step 14.1.1

Factor $4$ out of $8 \log_{a} (m)$ .

$\frac{1}{4} (4 (2 \log_{a} (m))) + \frac{1}{4} (12 \log_{a} (n)) + \frac{1}{4} \cdot - 3 + \frac{1}{4} (- 5 \log_{a} (b))$

Step 14.1.2

Cancel the common factor.

$\frac{1}{4} (4 (2 \log_{a} (m))) + \frac{1}{4} (12 \log_{a} (n)) + \frac{1}{4} \cdot - 3 + \frac{1}{4} (- 5 \log_{a} (b))$

Step 14.1.3

Rewrite the expression.

$2 \log_{a} (m) + \frac{1}{4} (12 \log_{a} (n)) + \frac{1}{4} \cdot - 3 + \frac{1}{4} (- 5 \log_{a} (b))$

Step 14.2

Cancel the common factor of $4$ .

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Step 14.2.1

Factor $4$ out of $12 \log_{a} (n)$ .

$2 \log_{a} (m) + \frac{1}{4} (4 (3 \log_{a} (n))) + \frac{1}{4} \cdot - 3 + \frac{1}{4} (- 5 \log_{a} (b))$

Step 14.2.2

Cancel the common factor.

$2 \log_{a} (m) + \frac{1}{4} (4 (3 \log_{a} (n))) + \frac{1}{4} \cdot - 3 + \frac{1}{4} (- 5 \log_{a} (b))$

Step 14.2.3

Rewrite the expression.

$2 \log_{a} (m) + 3 \log_{a} (n) + \frac{1}{4} \cdot - 3 + \frac{1}{4} (- 5 \log_{a} (b))$

Step 14.3

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

$2 \log_{a} (m) + 3 \log_{a} (n) + \frac{- 3}{4} + \frac{1}{4} (- 5 \log_{a} (b))$

Step 14.4

Multiply $\frac{1}{4} (- 5 \log_{a} (b))$ .

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Step 14.4.1

Combine $- 5$ and $\frac{1}{4}$ .

$2 \log_{a} (m) + 3 \log_{a} (n) + \frac{- 3}{4} + \frac{- 5}{4} \log_{a} (b)$

Step 14.4.2

Combine $\frac{- 5}{4}$ and $\log_{a} (b)$ .

$2 \log_{a} (m) + 3 \log_{a} (n) + \frac{- 3}{4} + \frac{- 5 \log_{a} (b)}{4}$

Step 15

Simplify each term.

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Step 15.1

Move the negative in front of the fraction.

$2 \log_{a} (m) + 3 \log_{a} (n) - \frac{3}{4} + \frac{- 5 \log_{a} (b)}{4}$

Step 15.2

Move the negative in front of the fraction.

$2 \log_{a} (m) + 3 \log_{a} (n) - \frac{3}{4} - \frac{5 \log_{a} (b)}{4}$