Precalculus Examples

$\log_{0.5} (\sqrt[4]{\frac{8}{2}})$

Step 1

Divide $8$ by $2$ .

$\log_{0.5} (\sqrt[4]{4})$

Step 2

Rewrite $4$ as $2^{2}$ .

$\log_{0.5} (\sqrt[4]{2^{2}})$

Step 3

Rewrite $\sqrt[4]{2^{2}}$ as $\sqrt{\sqrt{2^{2}}}$ .

$\log_{0.5} (\sqrt{\sqrt{2^{2}}})$

Step 4

Pull terms out from under the radical, assuming positive real numbers.

$\log_{0.5} (\sqrt{2})$

Step 5

Rewrite $\log_{0.5} (\sqrt{2})$ using the change of base formula.

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Step 5.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 5.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (\sqrt{2})}{\log (0.5)}$

Step 6

Log base $10$ of $0.5$ is approximately $- 0.30102999$ .

$\frac{\log (\sqrt{2})}{- 0.30102999}$

Step 7

Move the negative in front of the fraction.

$- \frac{\log (\sqrt{2})}{0.30102999}$

Step 8

Evaluate the root.

$- \frac{\log (1.41421356)}{0.30102999}$

Step 9

Log base $10$ of $1.41421356$ is approximately $0.15051499$ .

$- \frac{0.15051499}{0.30102999}$

Step 10

Divide $0.15051499$ by $0.30102999$ .

$- 1 \cdot 0.5$

Step 11

Multiply $- 1$ by $0.5$ .

$- 0.5$