Precalculus Examples

$\frac{\log_{4} (3)}{\log_{4} (2)}$

Step 1

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\log_{4} (3) \cdot 2$

Step 3

Multiply $\log_{4} (3) \cdot 2$ .

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Step 3.1

Reorder $\log_{4} (3)$ and $2$ .

$2 \cdot \log_{4} (3)$

Step 3.2

Simplify $2 \cdot \log_{4} (3)$ by moving $2$ inside the logarithm.

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

Step 4

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

$\log_{4} (9)$

Step 5

Rewrite $\log_{4} (9)$ 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 $b = 10$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Cancel the common factor of $2$ .

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Step 10.1

Cancel the common factor.

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

Step 10.2

Rewrite the expression.

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

Step 11

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.58496250 \dots$