Precalculus Examples

$\log_{9} ({(\frac{3^{3}}{7})}^{4})$

Step 1

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

$\log_{9} ({(\frac{27}{7})}^{4})$

Step 2

Apply the product rule to $\frac{27}{7}$ .

$\log_{9} (\frac{27^{4}}{7^{4}})$

Step 3

Raise $27$ to the power of $4$ .

$\log_{9} (\frac{531441}{7^{4}})$

Step 4

Raise $7$ to the power of $4$ .

$\log_{9} (\frac{531441}{2401})$

Step 5

Rewrite $\log_{9} (\frac{531441}{2401})$ 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 $b = 10$ .

$\frac{\log (\frac{531441}{2401})}{\log (9)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\frac{531441}{2401})}{\log (9)}$

Decimal Form:

$2.45751250 \dots$