Precalculus Examples

$\log_{6} (\sqrt{a \cdot b \cdot c})$

Step 1

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

$\log_{6} ({(a \cdot b \cdot c)}^{\frac{1}{2}})$

Step 2

Expand $\log_{6} ({(a \cdot b \cdot c)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{6} (a \cdot b \cdot c)$

Step 3

Rewrite $\log_{6} (a \cdot b \cdot c)$ as $\log_{6} (a \cdot b) + \log_{6} (c)$ .

$\frac{1}{2} (\log_{6} (a \cdot b) + \log_{6} (c))$

Step 4

Rewrite $\log_{6} (a \cdot b)$ as $\log_{6} (a) + \log_{6} (b)$ .

$\frac{1}{2} (\log_{6} (a) + \log_{6} (b) + \log_{6} (c))$

Step 5

Apply the distributive property.

$\frac{1}{2} \log_{6} (a) + \frac{1}{2} \log_{6} (b) + \frac{1}{2} \log_{6} (c)$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{2}$ and $\log_{6} (a)$ .

$\frac{\log_{6} (a)}{2} + \frac{1}{2} \log_{6} (b) + \frac{1}{2} \log_{6} (c)$

Step 6.2

Combine $\frac{1}{2}$ and $\log_{6} (b)$ .

$\frac{\log_{6} (a)}{2} + \frac{\log_{6} (b)}{2} + \frac{1}{2} \log_{6} (c)$

Step 6.3

Combine $\frac{1}{2}$ and $\log_{6} (c)$ .

$\frac{\log_{6} (a)}{2} + \frac{\log_{6} (b)}{2} + \frac{\log_{6} (c)}{2}$