Precalculus Examples

$\log_{13} ({(\frac{1}{3})}^{4})$

Step 1

Apply the product rule to $\frac{1}{3}$ .

$\log_{13} (\frac{1^{4}}{3^{4}})$

Step 2

One to any power is one.

$\log_{13} (\frac{1}{3^{4}})$

Step 3

Raise $3$ to the power of $4$ .

$\log_{13} (\frac{1}{81})$

Step 4

Rewrite $\log_{13} (\frac{1}{81})$ using the change of base formula.

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Step 4.1

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 4.2

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

$\frac{\log (\frac{1}{81})}{\log (13)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\frac{1}{81})}{\log (13)}$

Decimal Form:

$- 1.71326936 \dots$