Precalculus Examples

$\log (3 \sqrt{\frac{{MN}^{2}}{P}})$

Step 1

Rewrite $\log (3 \sqrt{\frac{{MN}^{2}}{P}})$ as $\log (3) + \log (\sqrt{\frac{{MN}^{2}}{P}})$ .

$\log (3) + \log (\sqrt{\frac{{MN}^{2}}{P}})$

Step 2

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

$\log (3) + \log ({(\frac{{MN}^{2}}{P})}^{\frac{1}{2}})$

Step 3

Expand $\log ({(\frac{{MN}^{2}}{P})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log (3) + \frac{1}{2} \log (\frac{{MN}^{2}}{P})$

Step 4

Rewrite $\log (\frac{{MN}^{2}}{P})$ as $\log ({MN}^{2}) - \log (P)$ .

$\log (3) + \frac{1}{2} (\log ({MN}^{2}) - \log (P))$

Step 5

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

$\log (3) + \frac{1}{2} (2 \log (MN) - \log (P))$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$\log (3) + \frac{1}{2} (2 \log (MN)) + \frac{1}{2} (- \log (P))$

Step 6.2

Cancel the common factor of $2$ .

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Step 6.2.1

Factor $2$ out of $2 \log (MN)$ .

$\log (3) + \frac{1}{2} (2 (\log (MN))) + \frac{1}{2} (- \log (P))$

Step 6.2.2

Cancel the common factor.

$\log (3) + \frac{1}{2} (2 \log (MN)) + \frac{1}{2} (- \log (P))$

Step 6.2.3

Rewrite the expression.

$\log (3) + \log (MN) + \frac{1}{2} (- \log (P))$

Step 6.3

Combine $\frac{1}{2}$ and $\log (P)$ .

$\log (3) + \log (MN) - \frac{\log (P)}{2}$