Precalculus Examples

$\log_{4} (3) \cdot \log_{27} (8)$

Step 1

Rewrite $\log_{4} (3)$ using the change of base formula.

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Step 1.1

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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{27} (8)$

Step 1.2

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

$\frac{\log (3)}{\log (4)} \log_{27} (8)$

Step 2

Rewrite $\log_{27} (8)$ using the change of base formula.

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Step 2.1

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 $0$ .

$\frac{\log (3)}{\log (4)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 2.2

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

$\frac{\log (3)}{\log (4)} \cdot \frac{\log (8)}{\log (27)}$

Step 3

Rewrite $\log (8)$ as $\log (2^{3})$ .

$\frac{\log (3)}{\log (4)} \cdot \frac{\log (2^{3})}{\log (27)}$

Step 4

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

$\frac{\log (3)}{\log (4)} \cdot \frac{3 \log (2)}{\log (27)}$

Step 5

Rewrite $\log (27)$ as $\log (3^{3})$ .

$\frac{\log (3)}{\log (4)} \cdot \frac{3 \log (2)}{\log (3^{3})}$

Step 6

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

$\frac{\log (3)}{\log (4)} \cdot \frac{3 \log (2)}{3 \log (3)}$

Step 7

Cancel the common factor of $3$ .

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Step 7.1

Cancel the common factor.

$\frac{\log (3)}{\log (4)} \cdot \frac{3 \log (2)}{3 \log (3)}$

Step 7.2

Rewrite the expression.

$\frac{\log (3)}{\log (4)} \cdot \frac{\log (2)}{\log (3)}$

Step 8

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

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Step 8.1

Cancel the common factor.

$\frac{\log (3)}{\log (4)} \cdot \frac{\log (2)}{\log (3)}$

Step 8.2

Rewrite the expression.

$\frac{1}{\log (4)} \log (2)$

Step 9

Combine $\frac{1}{\log (4)}$ and $\log (2)$ .

$\frac{\log (2)}{\log (4)}$

Step 10

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

$\frac{\log (2)}{\log (2^{2})}$

Step 11

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

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

Step 12

Cancel the common factor of $\log (2)$ .

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Step 12.1

Cancel the common factor.

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

Step 12.2

Rewrite the expression.

$\frac{1}{2}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$