Precalculus Examples

$\log_{b} (\sqrt[4]{\frac{m^{16} n^{8}}{b^{7}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{\frac{m^{16} n^{8}}{b^{7}}}$ as ${(\frac{m^{16} n^{8}}{b^{7}})}^{\frac{1}{4}}$ .

$\log_{b} ({(\frac{m^{16} n^{8}}{b^{7}})}^{\frac{1}{4}})$

Step 2

Expand $\log_{b} ({(\frac{m^{16} n^{8}}{b^{7}})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \log_{b} (\frac{m^{16} n^{8}}{b^{7}})$

Step 3

Rewrite $\log_{b} (\frac{m^{16} n^{8}}{b^{7}})$ as $\log_{b} (m^{16} n^{8}) - \log_{b} (b^{7})$ .

$\frac{1}{4} (\log_{b} (m^{16} n^{8}) - \log_{b} (b^{7}))$

Step 4

Expand $\log_{b} (b^{7})$ by moving $7$ outside the logarithm.

$\frac{1}{4} (\log_{b} (m^{16} n^{8}) - (7 \log_{b} (b)))$

Step 5

Logarithm base $b$ of $b$ is $1$ .

$\frac{1}{4} (\log_{b} (m^{16} n^{8}) - (7 \cdot 1))$

Step 6

Multiply $7$ by $1$ .

$\frac{1}{4} (\log_{b} (m^{16} n^{8}) - 1 \cdot 7)$

Step 7

Multiply $- 1$ by $7$ .

$\frac{1}{4} (\log_{b} (m^{16} n^{8}) - 7)$

Step 8

Rewrite $\log_{b} (m^{16} n^{8})$ as $\log_{b} (m^{16}) + \log_{b} (n^{8})$ .

$\frac{1}{4} (\log_{b} (m^{16}) + \log_{b} (n^{8}) - 7)$

Step 9

Expand $\log_{b} (m^{16})$ by moving $16$ outside the logarithm.

$\frac{1}{4} (16 \log_{b} (m) + \log_{b} (n^{8}) - 7)$

Step 10

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

$\frac{1}{4} (16 \log_{b} (m) + 8 \log_{b} (n) - 7)$

Step 11

Apply the distributive property.

$\frac{1}{4} (16 \log_{b} (m)) + \frac{1}{4} (8 \log_{b} (n)) + \frac{1}{4} \cdot - 7$

Step 12

Simplify.

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Step 12.1

Cancel the common factor of $4$ .

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Step 12.1.1

Factor $4$ out of $16 \log_{b} (m)$ .

$\frac{1}{4} (4 (4 \log_{b} (m))) + \frac{1}{4} (8 \log_{b} (n)) + \frac{1}{4} \cdot - 7$

Step 12.1.2

Cancel the common factor.

$\frac{1}{4} (4 (4 \log_{b} (m))) + \frac{1}{4} (8 \log_{b} (n)) + \frac{1}{4} \cdot - 7$

Step 12.1.3

Rewrite the expression.

$4 \log_{b} (m) + \frac{1}{4} (8 \log_{b} (n)) + \frac{1}{4} \cdot - 7$

Step 12.2

Cancel the common factor of $4$ .

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Step 12.2.1

Factor $4$ out of $8 \log_{b} (n)$ .

$4 \log_{b} (m) + \frac{1}{4} (4 (2 \log_{b} (n))) + \frac{1}{4} \cdot - 7$

Step 12.2.2

Cancel the common factor.

$4 \log_{b} (m) + \frac{1}{4} (4 (2 \log_{b} (n))) + \frac{1}{4} \cdot - 7$

Step 12.2.3

Rewrite the expression.

$4 \log_{b} (m) + 2 \log_{b} (n) + \frac{1}{4} \cdot - 7$

Step 12.3

Combine $\frac{1}{4}$ and $- 7$ .

$4 \log_{b} (m) + 2 \log_{b} (n) + \frac{- 7}{4}$

Step 13

Move the negative in front of the fraction.

$4 \log_{b} (m) + 2 \log_{b} (n) - \frac{7}{4}$