Precalculus Examples

$\log_{(5 - x)} (x - 3)$

Step 1

Set the base in $\log_{5 - x} (x - 3)$ greater than $0$ to find where the expression is defined.

$5 - x > 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $5$ from both sides of the inequality.

$- x > - 5$

Step 2.2

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

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Step 2.2.1

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 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2.2

Divide $x$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 5$ by $- 1$ .

$x < 5$

Step 3

Set the argument in $\log_{5 - x} (x - 3)$ greater than $0$ to find where the expression is defined.

$x - 3 > 0$

Step 4

Add $3$ to both sides of the inequality.

$x > 3$

Step 5

Set the base in $\log_{5 - x} (x - 3)$ equal to $1$ to find where the expression is undefined.

$5 - x = 1$

Step 6

Solve for $x$ .

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Step 6.1

Move all terms not containing $x$ to the right side of the equation.

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Step 6.1.1

Subtract $5$ from both sides of the equation.

$- x = 1 - 5$

Step 6.1.2

Subtract $5$ from $1$ .

$- x = - 4$

Step 6.2

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 6.2.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 6.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 7

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

Interval Notation:

$(3, 4) \cup (4, 5)$

Set-Builder Notation:

${x | 3 < x < 5, x \neq 4}$

Step 8