Precalculus Examples

$\log_{4} (x - 5) - 1$

Step 1

Rewrite $\log_{4} (x - 5)$ 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 $0$ .

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

Step 1.2

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

$\frac{\log (x - 5)}{\log (4)} - 1$

Step 2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{\log (4)}{\log (4)}$ .

$\frac{\log (x - 5)}{\log (4)} - 1 \cdot \frac{\log (4)}{\log (4)}$

Step 3

Combine $- 1$ and $\frac{\log (4)}{\log (4)}$ .

$\frac{\log (x - 5)}{\log (4)} + \frac{- \log (4)}{\log (4)}$

Step 4

Combine the numerators over the common denominator.

$\frac{\log (x - 5) - \log (4)}{\log (4)}$

Step 5

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

$\frac{\log (\frac{x - 5}{4})}{\log (4)}$