Calculus Examples

$\frac{x}{1 - \ln (x - 9)}$

Step 1

Set the argument in $\ln (x - 9)$ greater than $0$ to find where the expression is defined.

$x - 9 > 0$

Step 2

Add $9$ to both sides of the inequality.

$x > 9$

Step 3

Set the denominator in $\frac{x}{1 - \ln (x - 9)}$ equal to $0$ to find where the expression is undefined.

$1 - \ln (x - 9) = 0$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $1$ from both sides of the equation.

$- \ln (x - 9) = - 1$

Step 4.2

Divide each term in $- \ln (x - 9) = - 1$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- \ln (x - 9) = - 1$ by $- 1$ .

$\frac{- \ln (x - 9)}{- 1} = \frac{- 1}{- 1}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{\ln (x - 9)}{1} = \frac{- 1}{- 1}$

Step 4.2.2.2

Divide $\ln (x - 9)$ by $1$ .

$\ln (x - 9) = \frac{- 1}{- 1}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 1$ by $- 1$ .

$\ln (x - 9) = 1$

Step 4.3

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

$e^{\ln (x - 9)} = e^{1}$

Step 4.4

Rewrite $\ln (x - 9) = 1$ 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 $b^{y} = x$ .

$e^{1} = x - 9$

Step 4.5

Solve for $x$ .

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Step 4.5.1

Rewrite the equation as $x - 9 = e^{1}$ .

$x - 9 = e$

Step 4.5.2

Add $9$ to both sides of the equation.

$x = e + 9$

Step 5

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

Interval Notation:

$(9, e + 9) \cup (e + 9, \infty)$

Set-Builder Notation:

${x | x > 9, x \neq e + 9}$

Step 6