Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

$\frac{\log_{2} (9) - 1}{2 \log_{2} (3)}$

Step 2

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

$\frac{\log_{2} (9) - 1}{\log_{2} (3^{2})}$

Step 3

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

$\frac{\log_{2} (9) - 1}{\log_{2} (9)}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $\log_{2} (9)$ using the change of base formula.

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Step 4.1.1

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 $0$ .

$\frac{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} - 1}{\log_{2} (9)}$

Step 4.1.2

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

$\frac{\frac{\log (9)}{\log (2)} - 1}{\log_{2} (9)}$

Step 4.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{\log (2)}{\log (2)}$ .

$\frac{\frac{\log (9)}{\log (2)} - 1 \cdot \frac{\log (2)}{\log (2)}}{\log_{2} (9)}$

Step 4.3

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

$\frac{\frac{\log (9)}{\log (2)} + \frac{- \log (2)}{\log (2)}}{\log_{2} (9)}$

Step 4.4

Combine the numerators over the common denominator.

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

Step 4.5

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{\frac{\log (\frac{9}{2})}{\log (2)}}{\log_{2} (9)}$

Step 5

Rewrite $\log_{2} (9)$ 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 $0$ .

$\frac{\frac{\log (\frac{9}{2})}{\log (2)}}{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}$

Step 5.2

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

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

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log (\frac{9}{2})}{\log (2)} \cdot \frac{\log (2)}{\log (9)}$

Step 7

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

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Step 7.1

Cancel the common factor.

$\frac{\log (\frac{9}{2})}{\log (2)} \cdot \frac{\log (2)}{\log (9)}$

Step 7.2

Rewrite the expression.

$\log (\frac{9}{2}) \frac{1}{\log (9)}$

Step 8

Combine $\log (\frac{9}{2})$ and $\frac{1}{\log (9)}$ .

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.68453512 \dots$