Precalculus Examples

$\log_{m} (\sqrt{\frac{{(n + 2)}^{4}}{m}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{{(n + 2)}^{4}}{m}}$ as ${(\frac{{(n + 2)}^{4}}{m})}^{\frac{1}{2}}$ .

$\log_{m} ({(\frac{{(n + 2)}^{4}}{m})}^{\frac{1}{2}})$

Step 2

Expand $\log_{m} ({(\frac{{(n + 2)}^{4}}{m})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{m} (\frac{{(n + 2)}^{4}}{m})$

Step 3

Rewrite $\log_{m} (\frac{{(n + 2)}^{4}}{m})$ as $\log_{m} ({(n + 2)}^{4}) - \log_{m} (m)$ .

$\frac{1}{2} (\log_{m} ({(n + 2)}^{4}) - \log_{m} (m))$

Step 4

Expand $\log_{m} ({(n + 2)}^{4})$ by moving $4$ outside the logarithm.

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

Step 5

Logarithm base $m$ of $m$ is $1$ .

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

Step 6

Multiply $- 1$ by $1$ .

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

Step 7

Apply the distributive property.

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

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Factor $2$ out of $4 \log_{m} (n + 2)$ .

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

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

Step 9

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

$2 \log_{m} (n + 2) + \frac{- 1}{2}$

Step 10

Move the negative in front of the fraction.

$2 \log_{m} (n + 2) - \frac{1}{2}$

Step 11

To write $2 \log_{m} (n + 2)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 12

Combine $2 \log_{m} (n + 2)$ and $\frac{2}{2}$ .

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

Step 13

Combine the numerators over the common denominator.

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

Step 14

Multiply $2$ by $2$ .

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