Precalculus Examples

$\log_{4} (3^{2}) (\log_{3} (4^{- 3}))$

Step 1

Raise $3$ to the power of $2$ .

$\log_{4} (9) \log_{3} (4^{- 3})$

Step 2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\log_{4} (9) \log_{3} (\frac{1}{4^{3}})$

Step 3

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

$\log_{4} (9) \log_{3} (\frac{1}{64})$

Step 4

Rewrite $\log_{4} (9)$ using the change of base formula.

Tap for more steps...

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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{3} (\frac{1}{64})$

Step 4.2

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

$\frac{\log (9)}{\log (4)} \log_{3} (\frac{1}{64})$

Step 5

Rewrite $\log (9)$ as $\log (3^{2})$ .

$\frac{\log (3^{2})}{\log (4)} \log_{3} (\frac{1}{64})$

Step 6

Expand $\log (3^{2})$ by moving $2$ outside the logarithm.

$\frac{2 \log (3)}{\log (4)} \log_{3} (\frac{1}{64})$

Step 7

Rewrite $\log (4)$ as $\log (2^{2})$ .

$\frac{2 \log (3)}{\log (2^{2})} \log_{3} (\frac{1}{64})$

Step 8

Expand $\log (2^{2})$ by moving $2$ outside the logarithm.

$\frac{2 \log (3)}{2 \log (2)} \log_{3} (\frac{1}{64})$

Step 9

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.1

Cancel the common factor.

$\frac{2 \log (3)}{2 \log (2)} \log_{3} (\frac{1}{64})$

Step 9.2

Rewrite the expression.

$\frac{\log (3)}{\log (2)} \log_{3} (\frac{1}{64})$

Step 10

Rewrite $\log_{3} (\frac{1}{64})$ using the change of base formula.

Tap for more steps...

Step 10.1

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 $0$ .

$\frac{\log (3)}{\log (2)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 10.2

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

$\frac{\log (3)}{\log (2)} \cdot \frac{\log (\frac{1}{64})}{\log (3)}$

Step 11

Cancel the common factor of $\log (3)$ .

Tap for more steps...

Step 11.1

Cancel the common factor.

$\frac{\log (3)}{\log (2)} \cdot \frac{\log (\frac{1}{64})}{\log (3)}$

Step 11.2

Rewrite the expression.

$\frac{1}{\log (2)} \log (\frac{1}{64})$

Step 12

Combine $\frac{1}{\log (2)}$ and $\log (\frac{1}{64})$ .

$\frac{\log (\frac{1}{64})}{\log (2)}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\frac{1}{64})}{\log (2)}$

Decimal Form:

$- 6$