Precalculus Examples

$\frac{\log_{3} (20)}{2}$

Step 1

Rewrite $\frac{\log_{3} (20)}{2}$ as $\frac{1}{2} \log_{3} (20)$ .

$\frac{1}{2} \log_{3} (20)$

Step 2

Simplify $\frac{1}{2} \log_{3} (20)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{3} (20^{\frac{1}{2}})$

Step 3

Rewrite $\log_{3} (20^{\frac{1}{2}})$ using the change of base formula.

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Step 3.1

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 3.2

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

$\frac{\log (20^{\frac{1}{2}})}{\log (3)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (20^{\frac{1}{2}})}{\log (3)}$

Decimal Form:

$1.36341651 \dots$