Precalculus Examples

$f (x) = \frac{4}{1 - 2 \ln (x)}$

Step 1

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

$x > 0$

Step 2

Set the denominator in $\frac{4}{1 - 2 \ln (x)}$ equal to $0$ to find where the expression is undefined.

$1 - 2 \ln (x) = 0$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $1$ from both sides of the equation.

$- 2 \ln (x) = - 1$

Step 3.2

Divide each term in $- 2 \ln (x) = - 1$ by $- 2$ and simplify.

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Step 3.2.1

Divide each term in $- 2 \ln (x) = - 1$ by $- 2$ .

$\frac{- 2 \ln (x)}{- 2} = \frac{- 1}{- 2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 2 \ln (x)}{- 2} = \frac{- 1}{- 2}$

Step 3.2.2.1.2

Divide $\ln (x)$ by $1$ .

$\ln (x) = \frac{- 1}{- 2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Dividing two negative values results in a positive value.

$\ln (x) = \frac{1}{2}$

Step 3.3

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

$e^{\ln (x)} = e^{\frac{1}{2}}$

Step 3.4

Rewrite $\ln (x) = \frac{1}{2}$ 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 $b^{y} = x$ .

$e^{\frac{1}{2}} = x$

Step 3.5

Rewrite the equation as $x = e^{\frac{1}{2}}$ .

$x = e^{\frac{1}{2}}$

Step 4

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

Interval Notation:

$(0, e^{\frac{1}{2}}) \cup (e^{\frac{1}{2}}, \infty)$

Set-Builder Notation:

${x | x > 0, x \neq e^{\frac{1}{2}}}$

Step 5