Precalculus Examples

$\log_{5} (x - 1) + \log_{5} (x - 2) - \log_{5} (x + 6)$

Step 1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{5} ((x - 1) (x - 2)) - \log_{5} (x + 6)$

Step 2

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

$\log_{5} (\frac{(x - 1) (x - 2)}{x + 6})$

Step 3

Rewrite $\log_{5} (\frac{(x - 1) (x - 2)}{x + 6})$ using the change of base formula.

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Step 3.1

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 3.2

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

$\frac{\log (\frac{(x - 1) (x - 2)}{x + 6})}{\log (5)}$