Precalculus Examples

$\log_{4} (11) \log_{4} (9)$

Step 1

Rewrite $\log_{4} (11)$ 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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{4} (9)$

Step 1.2

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

$\frac{\log (11)}{\log (4)} \log_{4} (9)$

Step 2

Rewrite $\log_{4} (9)$ 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 $0$ .

$\frac{\log (11)}{\log (4)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 2.2

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

$\frac{\log (11)}{\log (4)} \cdot \frac{\log (9)}{\log (4)}$

Step 3

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

$\frac{\log (11)}{\log (4)} \cdot \frac{\log (3^{2})}{\log (4)}$

Step 4

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

$\frac{\log (11)}{\log (4)} \cdot \frac{2 \log (3)}{\log (4)}$

Step 5

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

$\frac{\log (11)}{\log (4)} \cdot \frac{2 \log (3)}{\log (2^{2})}$

Step 6

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

$\frac{\log (11)}{\log (4)} \cdot \frac{2 \log (3)}{2 \log (2)}$

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Cancel the common factor.

$\frac{\log (11)}{\log (4)} \cdot \frac{2 \log (3)}{2 \log (2)}$

Step 7.2

Rewrite the expression.

$\frac{\log (11)}{\log (4)} \cdot \frac{\log (3)}{\log (2)}$

Step 8

Multiply $\frac{\log (11)}{\log (4)}$ by $\frac{\log (3)}{\log (2)}$ .

$\frac{\log (11) \log (3)}{\log (4) \log (2)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (11) \log (3)}{\log (4) \log (2)}$

Decimal Form:

$2.74153469 \dots$