Precalculus Examples

$\log_{3} (e) - \log ({(e)}^{\ln (x - 2)})$

Step 1

Exponentiation and log are inverse functions.

$\log_{3} (e) - \log (x - 2)$

Step 2

Rewrite $\log_{3} (e)$ using the change of base formula.

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Step 2.1

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 $0$ .

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

Step 2.2

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

$\frac{\log (e)}{\log (3)} - \log (x - 2)$

Step 3

To write $- \log (x - 2)$ as a fraction with a common denominator, multiply by $\frac{\log (3)}{\log (3)}$ .

$\frac{\log (e)}{\log (3)} - \log (x - 2) \cdot \frac{\log (3)}{\log (3)}$

Step 4

Combine $- \log (x - 2)$ and $\frac{\log (3)}{\log (3)}$ .

$\frac{\log (e)}{\log (3)} + \frac{- \log (x - 2) \log (3)}{\log (3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{\log (e) - \log (x - 2) \log (3)}{\log (3)}$