Calculus Examples

$f (x) = \ln (e^{x} - 8)$

Step 1

Set the argument in $\ln (e^{x} - 8)$ greater than $0$ to find where the expression is defined.

$e^{x} - 8 > 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $8$ to both sides of the inequality.

$e^{x} > 8$

Step 2.2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x}) > \ln (8)$

Step 2.3

Expand the left side.

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Step 2.3.1

Expand $\ln (e^{x})$ by moving $x$ outside the logarithm.

$x \ln (e) > \ln (8)$

Step 2.3.2

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

$x \cdot 1 > \ln (8)$

Step 2.3.3

Multiply $x$ by $1$ .

$x > \ln (8)$

Step 3

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

Interval Notation:

$(\ln (8), \infty)$

Set-Builder Notation:

${x | x > \ln (8)}$

Step 4