Precalculus Examples

$\log_{4} (z \sqrt{x y})$

Step 1

Rewrite $\log_{4} (z \sqrt{x y})$ as $\log_{4} (z) + \log_{4} (\sqrt{x y})$ .

$\log_{4} (z) + \log_{4} (\sqrt{x y})$

Step 2

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

$\log_{4} (z) + \log_{4} ({(x y)}^{\frac{1}{2}})$

Step 3

Expand $\log_{4} ({(x y)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{4} (z) + \frac{1}{2} \log_{4} (x y)$

Step 4

Rewrite $\log_{4} (x y)$ as $\log_{4} (x) + \log_{4} (y)$ .

$\log_{4} (z) + \frac{1}{2} (\log_{4} (x) + \log_{4} (y))$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

$\log_{4} (z) + \frac{1}{2} \log_{4} (x) + \frac{1}{2} \log_{4} (y)$

Step 5.2

Combine $\frac{1}{2}$ and $\log_{4} (x)$ .

$\log_{4} (z) + \frac{\log_{4} (x)}{2} + \frac{1}{2} \log_{4} (y)$

Step 5.3

Combine $\frac{1}{2}$ and $\log_{4} (y)$ .

$\log_{4} (z) + \frac{\log_{4} (x)}{2} + \frac{\log_{4} (y)}{2}$