Precalculus Examples

$\log_{m} (\sqrt{\frac{A^{3} B}{C}})$

Step 1

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

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

Step 2

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

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

Step 3

Rewrite $\log_{m} (\frac{A^{3} B}{C})$ as $\log_{m} (A^{3} B) - \log_{m} (C)$ .

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

Step 4

Rewrite $\log_{m} (A^{3} B)$ as $\log_{m} (A^{3}) + \log_{m} (B)$ .

$\frac{1}{2} (\log_{m} (A^{3}) + \log_{m} (B) - \log_{m} (C))$

Step 5

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

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

Step 6

Apply the distributive property.

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

Step 7

Simplify.

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Step 7.1

Multiply $\frac{1}{2} (3 \log_{m} (A))$ .

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Step 7.1.1

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

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

Step 7.1.2

Combine $\frac{3}{2}$ and $\log_{m} (A)$ .

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

Step 7.2

Combine $\frac{1}{2}$ and $\log_{m} (B)$ .

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

Step 7.3

Combine $\frac{1}{2}$ and $\log_{m} (C)$ .

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