Precalculus Examples

$\log_{2} (4) - \log_{4} (9)$

Step 1

Logarithm base $2$ of $4$ is $2$ .

$2 - \log_{4} (9)$

Step 2

Simplify each term.

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Step 2.1

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

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Step 2.1.1

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 $0$ .

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

Step 2.1.2

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

$2 - \frac{\log (9)}{\log (4)}$

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

$2 - \frac{2 \log (3)}{\log (2^{2})}$

Step 2.5

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

$2 - \frac{2 \log (3)}{2 \log (2)}$

Step 2.6

Cancel the common factor of $2$ .

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Step 2.6.1

Cancel the common factor.

$2 - \frac{2 \log (3)}{2 \log (2)}$

Step 2.6.2

Rewrite the expression.

$2 - \frac{\log (3)}{\log (2)}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$2 - \frac{\log (3)}{\log (2)}$

Decimal Form:

$0.41503749 \dots$