Precalculus Examples

$\log_{b} (\sqrt[3]{\frac{M^{3}}{N^{2}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{M^{3}}{N^{2}}}$ as ${(\frac{M^{3}}{N^{2}})}^{\frac{1}{3}}$ .

$\log_{b} ({(\frac{M^{3}}{N^{2}})}^{\frac{1}{3}})$

Step 2

Expand $\log_{b} ({(\frac{M^{3}}{N^{2}})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{b} (\frac{M^{3}}{N^{2}})$

Step 3

Rewrite $\log_{b} (\frac{M^{3}}{N^{2}})$ as $\log_{b} (M^{3}) - \log_{b} (N^{2})$ .

$\frac{1}{3} (\log_{b} (M^{3}) - \log_{b} (N^{2}))$

Step 4

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

$\frac{1}{3} (3 \log_{b} (M) - \log_{b} (N^{2}))$

Step 5

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

$\frac{1}{3} (3 \log_{b} (M) - (2 \log_{b} (N)))$

Step 6

Multiply $2$ by $- 1$ .

$\frac{1}{3} (3 \log_{b} (M) - 2 \log_{b} (N))$

Step 7

Apply the distributive property.

$\frac{1}{3} (3 \log_{b} (M)) + \frac{1}{3} (- 2 \log_{b} (N))$

Step 8

Cancel the common factor of $3$ .

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Step 8.1

Factor $3$ out of $3 \log_{b} (M)$ .

$\frac{1}{3} (3 (\log_{b} (M))) + \frac{1}{3} (- 2 \log_{b} (N))$

Step 8.2

Cancel the common factor.

$\frac{1}{3} (3 \log_{b} (M)) + \frac{1}{3} (- 2 \log_{b} (N))$

Step 8.3

Rewrite the expression.

$\log_{b} (M) + \frac{1}{3} (- 2 \log_{b} (N))$

Step 9

Multiply $\frac{1}{3} (- 2 \log_{b} (N))$ .

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Step 9.1

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

$\log_{b} (M) + \frac{- 2}{3} \log_{b} (N)$

Step 9.2

Combine $\frac{- 2}{3}$ and $\log_{b} (N)$ .

$\log_{b} (M) + \frac{- 2 \log_{b} (N)}{3}$

Step 10

Move the negative in front of the fraction.

$\log_{b} (M) - \frac{2 \log_{b} (N)}{3}$