Graph y=| natural log of x|

$y = | \ln (x) |$

Find the absolute value vertex. In this case, the vertex for $y = | \ln (x) |$ is $(1, 0)$ .

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To find the $x$ coordinate of the vertex, set the inside of the absolute value $\ln (x)$ equal to $0$ . In this case, $\ln (x) = 0$ .

$\ln (x) = 0$

Solve the equation $\ln (x) = 0$ to find the $x$ coordinate for the absolute value vertex.

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

$e^{\ln (x)} = e^{0}$

Rewrite $\ln (x) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{0} = x$

Solve for $x$ .

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Rewrite the equation as $x = e^{0}$ .

$x = e^{0}$

Anything raised to $0$ is $1$ .

$x = 1$

Replace the variable $x$ with $1$ in the expression.

$y = | \ln (1) |$

Simplify $| \ln (1) |$ .

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The natural logarithm of $1$ is $0$ .

$y = | 0 |$

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

$y = 0$

The absolute value vertex is $(1, 0)$ .

$(1, 0)$

Find the domain for $y = | \ln (x) |$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the absolute value function.

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Set the argument in $\ln (x)$ greater than $0$ to find where the expression is defined.

$x > 0$

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

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

For each $x$ value, there is one $y$ value. Select few $x$ values from the domain. It would be more useful to select the values so that they are around the $x$ value of the absolute value vertex.

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Substitute the $x$ value $2$ into $f (x) = | \ln (x) |$ . In this case, the point is $(2, \ln (2))$ .

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Replace the variable $x$ with $2$ in the expression.

$f (2) = | \ln (2) |$

Simplify the result.

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$\ln (2)$ is approximately $0.69314718$ which is positive so remove the absolute value

$f (2) = \ln (2)$

The final answer is $\ln (2)$ .

$y = \ln (2)$

Substitute the $x$ value $3$ into $f (x) = | \ln (x) |$ . In this case, the point is $(3, \ln (3))$ .

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Replace the variable $x$ with $3$ in the expression.

$f (3) = | \ln (3) |$

Simplify the result.

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$\ln (3)$ is approximately $1.09861228$ which is positive so remove the absolute value

$f (3) = \ln (3)$

The final answer is $\ln (3)$ .

$y = \ln (3)$

The absolute value can be graphed using the points around the vertex $(1, 0), (2, 0.69), (3, 1.1)$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 0.69 \\ 3 & 1.1 \end{array}$

Graph y=| natural log of x|

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