Evaluate 2 log base 3 of 1/52

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

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

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

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

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

One to any power is one.

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

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

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

Rewrite $\log_{3} (\frac{1}{2704})$ using the change of base formula.

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 $0$ .

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

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

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 7.19315405 \dots$