Precalculus Examples

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

Step 1

Rewrite $\log_{4} (\frac{\sqrt{x}}{y z^{2}})$ as $\log_{4} (\sqrt{x}) - \log_{4} (y z^{2})$ .

$\log_{4} (\sqrt{x}) - \log_{4} (y z^{2})$

Step 2

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

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

Step 3

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

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

Step 4

Rewrite $\log_{4} (y z^{2})$ as $\log_{4} (y) + \log_{4} (z^{2})$ .

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

Step 5

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

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

Step 6

Simplify each term.

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Step 6.1

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $2$ by $- 1$ .

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