Precalculus Examples

$f (x) = \log_{x} (5 - x)$

Step 1

Set the base in $\log_{x} (5 - x)$ greater than $0$ to find where the expression is defined.

$x > 0$

Step 2

Set the argument in $\log_{x} (5 - x)$ greater than $0$ to find where the expression is defined.

$5 - x > 0$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $5$ from both sides of the inequality.

$- x > - 5$

Step 3.2

Divide each term in $- x > - 5$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- x > - 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{- 5}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{- 5}{- 1}$

Step 3.2.2.2

Divide $x$ by $1$ .

$x < \frac{- 5}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 5$ by $- 1$ .

$x < 5$

Step 4

Set the base in $\log_{x} (5 - x)$ equal to $1$ to find where the expression is undefined.

$x = 1$

Step 5

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(0, 1) \cup (1, 5)$

Set-Builder Notation:

${x | 0 < x < 5, x \neq 1}$

Step 6