To find the coordinate of the vertex, set the inside of the absolute value equal to . In this case, .

Solve the equation to find the coordinate for the absolute value vertex.

To solve for , rewrite the equation using properties of logarithms.

Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .

Solve for .

Rewrite the equation as .

Anything raised to is .

Replace the variable with in the expression.

Simplify .

The natural logarithm of is .

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

The absolute value vertex is .

Set the argument in greater than to find where the expression is defined.

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

Interval Notation:

Set-Builder Notation:

Interval Notation:

Set-Builder Notation:

Substitute the value into . In this case, the point is .

Replace the variable with in the expression.

Simplify the result.

is approximately which is positive so remove the absolute value

The final answer is .

Substitute the value into . In this case, the point is .

Replace the variable with in the expression.

Simplify the result.

is approximately which is positive so remove the absolute value

The final answer is .

The absolute value can be graphed using the points around the vertex

Graph y=| natural log of x|