Calculus Examples

$\log ({(\frac{A B}{C^{2}})}^{\frac{1}{3}})$

Step 1

Expand $\log ({(\frac{A B}{C^{2}})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log (\frac{A B}{C^{2}})$

Step 2

Rewrite $\log (\frac{A B}{C^{2}})$ as $\log (A B) - \log (C^{2})$ .

$\frac{1}{3} (\log (A B) - \log (C^{2}))$

Step 3

Expand $\log (C^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{3} (\log (A B) - (2 \log (C)))$

Step 4

Multiply $2$ by $- 1$ .

$\frac{1}{3} (\log (A B) - 2 \log (C))$

Step 5

Rewrite $\log (A B)$ as $\log (A) + \log (B)$ .

$\frac{1}{3} (\log (A) + \log (B) - 2 \log (C))$

Step 6

Apply the distributive property.

$\frac{1}{3} \log (A) + \frac{1}{3} \log (B) + \frac{1}{3} (- 2 \log (C))$

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{3}$ and $\log (A)$ .

$\frac{\log (A)}{3} + \frac{1}{3} \log (B) + \frac{1}{3} (- 2 \log (C))$

Step 7.2

Combine $\frac{1}{3}$ and $\log (B)$ .

$\frac{\log (A)}{3} + \frac{\log (B)}{3} + \frac{1}{3} (- 2 \log (C))$

Step 7.3

Multiply $\frac{1}{3} (- 2 \log (C))$ .

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Step 7.3.1

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

$\frac{\log (A)}{3} + \frac{\log (B)}{3} + \frac{- 2}{3} \log (C)$

Step 7.3.2

Combine $\frac{- 2}{3}$ and $\log (C)$ .

$\frac{\log (A)}{3} + \frac{\log (B)}{3} + \frac{- 2 \log (C)}{3}$

Step 8

Move the negative in front of the fraction.

$\frac{\log (A)}{3} + \frac{\log (B)}{3} - \frac{2 \log (C)}{3}$