Precalculus Examples

$\log_{24} (x - \frac{1}{2})$

Step 1

Rewrite $\log_{24} (x - \frac{1}{2})$ 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 $0$ .

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

Step 1.2

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

$\frac{\log (x - \frac{1}{2})}{\log (24)}$

Step 2

Simplify the numerator.

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Step 2.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\log (x \cdot \frac{2}{2} - \frac{1}{2})}{\log (24)}$

Step 2.2

Combine $x$ and $\frac{2}{2}$ .

$\frac{\log (\frac{x \cdot 2}{2} - \frac{1}{2})}{\log (24)}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{\log (\frac{x \cdot 2 - 1}{2})}{\log (24)}$

Step 2.4

Move $2$ to the left of $x$ .

$\frac{\log (\frac{2 x - 1}{2})}{\log (24)}$