Calculus Examples

$\frac{1}{3 - \ln (x)}$

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 $\frac{1}{3 - \ln (x)}$ equal to $0$ to find where the expression is undefined.

$3 - \ln (x) = 0$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $3$ from both sides of the equation.

$- \ln (x) = - 3$

Step 3.2

Divide each term in $- \ln (x) = - 3$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- \ln (x) = - 3$ by $- 1$ .

$\frac{- \ln (x)}{- 1} = \frac{- 3}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{\ln (x)}{1} = \frac{- 3}{- 1}$

Step 3.2.2.2

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

$\ln (x) = \frac{- 3}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 3$ by $- 1$ .

$\ln (x) = 3$

Step 3.3

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

$e^{\ln (x)} = e^{3}$

Step 3.4

Rewrite $\ln (x) = 3$ 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 $b^{y} = x$ .

$e^{3} = x$

Step 3.5

Rewrite the equation as $x = e^{3}$ .

$x = e^{3}$

Step 4

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

Interval Notation:

$(0, e^{3}) \cup (e^{3}, \infty)$

Set-Builder Notation:

${x | x > 0, x \neq e^{3}}$

Step 5