Algebra Examples

$2 \log_{3} (\frac{1}{52})$

Step 1

Simplify $2 \log_{3} (\frac{1}{52})$ by moving $2$ inside the logarithm.

$\log_{3} ({(\frac{1}{52})}^{2})$

Step 2

Apply the product rule to $\frac{1}{52}$ .

$\log_{3} (\frac{1^{2}}{52^{2}})$

Step 3

One to any power is one.

$\log_{3} (\frac{1}{52^{2}})$

Step 4

Raise $52$ to the power of $2$ .

$\log_{3} (\frac{1}{2704})$

Step 5

Rewrite $\log_{3} (\frac{1}{2704})$ 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 $b = 10$ .

$\frac{\log (\frac{1}{2704})}{\log (3)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\frac{1}{2704})}{\log (3)}$

Decimal Form:

$- 7.19315405 \dots$