Find the Domain natural log of (x-8)^2

$\ln ({(x - 8)}^{2})$

Step 1

Set the argument in $\ln ({(x - 8)}^{2})$ greater than $0$ to find where the expression is defined.

${(x - 8)}^{2} > 0$

Step 2

Solve for $x$ .

Take the square root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{{(x - 8)}^{2}} > \sqrt{0}$

Simplify the equation.

Simplify the left side.

Pull terms out from under the radical.

$| x - 8 | > \sqrt{0}$

Simplify the right side.

Simplify $\sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$| x - 8 | > \sqrt{0^{2}}$

Pull terms out from under the radical.

$| x - 8 | > | 0 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$| x - 8 | > 0$

Write $| x - 8 | > 0$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x - 8 \geq 0$

Add $8$ to both sides of the inequality.

$x \geq 8$

In the piece where $x - 8$ is non-negative, remove the absolute value.

$x - 8 > 0$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x - 8 < 0$

Add $8$ to both sides of the inequality.

$x < 8$

In the piece where $x - 8$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x - 8) > 0$

Write as a piecewise.

${\begin{cases} x - 8 > 0 & x \geq 8 \\ - (x - 8) > 0 & x < 8 \end{cases}$

Simplify $- (x - 8) > 0$ .

Apply the distributive property.

${\begin{cases} x - 8 > 0 & x \geq 8 \\ - x - - 8 > 0 & x < 8 \end{cases}$

Multiply $- 1$ by $- 8$ .

${\begin{cases} x - 8 > 0 & x \geq 8 \\ - x + 8 > 0 & x < 8 \end{cases}$

Add $8$ to both sides of the inequality.

$x > 8$

Solve $- x + 8 > 0$ for $x$ .

Subtract $8$ from both sides of the inequality.

$- x > - 8$

Divide each term in $- x > - 8$ by $- 1$ and simplify.

Divide each term in $- x > - 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{- 8}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{- 8}{- 1}$

Divide $x$ by $1$ .

$x < \frac{- 8}{- 1}$

Simplify the right side.

Divide $- 8$ by $- 1$ .

$x < 8$

Find the union of the solutions.

$x < 8$ or $x > 8$

Step 3

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

Interval Notation:

$(- \infty, 8) \cup (8, \infty)$

Set-Builder Notation:

${x | x \neq 8}$

Step 4

Find the Domain natural log of (x-8)^2

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