Algebra Examples

$\log_{3} (\sqrt{100})$

Step 1

Rewrite $100$ as $10^{2}$ .

$\log_{3} (\sqrt{10^{2}})$

Step 2

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

$\log_{3} (10)$

Step 3

Rewrite $\log_{3} (10)$ 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 $b = 10$ .

$\frac{\log (10)}{\log (3)}$

Step 4

Logarithm base $10$ of $10$ is $1$ .

$\frac{1}{\log (3)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{\log (3)}$

Decimal Form:

$2.09590327 \dots$