Finite Math Examples

$\ln (x^{4} e^{- x^{3}})$

Step 1

Set the argument in $\ln (x^{4} e^{- x^{3}})$ greater than $0$ to find where the expression is defined.

$x^{4} e^{- x^{3}} > 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{4} = 0$

$e^{- x^{3}} = 0$

Step 2.2

Set $x^{4}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.2.1

Set $x^{4}$ equal to $0$ .

$x^{4} = 0$

Step 2.2.2

Solve $x^{4} = 0$ for $x$ .

Tap for more steps...

Step 2.2.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{0}$

Step 2.2.2.2

Simplify $\pm \sqrt[4]{0}$ .

Tap for more steps...

Step 2.2.2.2.1

Rewrite $0$ as $0^{4}$ .

$x = \pm \sqrt[4]{0^{4}}$

Step 2.2.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 2.2.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.3

Set $e^{- x^{3}}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.3.1

Set $e^{- x^{3}}$ equal to $0$ .

$e^{- x^{3}} = 0$

Step 2.3.2

Solve $e^{- x^{3}} = 0$ for $x$ .

Tap for more steps...

Step 2.3.2.1

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

$\ln (e^{- x^{3}}) = \ln (0)$

Step 2.3.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 2.3.2.3

There is no solution for $e^{- x^{3}} = 0$

No solution

Step 2.4

The final solution is all the values that make $x^{4} e^{- x^{3}} > 0$ true.

$x = 0$

Step 2.5

Use each root to create test intervals.

$x < 0$

$x > 0$

Step 2.6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Tap for more steps...

Step 2.6.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

Tap for more steps...

Step 2.6.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 2.6.1.2

Replace $x$ with $- 2$ in the original inequality.

${(- 2)}^{4} e^{- {(- 2)}^{3}} > 0$

Step 2.6.1.3

The left side $47695.32779266$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.6.2

Test a value on the interval $x > 0$ to see if it makes the inequality true.

Tap for more steps...

Step 2.6.2.1

Choose a value on the interval $x > 0$ and see if this value makes the original inequality true.

$x = 2$

Step 2.6.2.2

Replace $x$ with $2$ in the original inequality.

${(2)}^{4} e^{- {(2)}^{3}} > 0$

Step 2.6.2.3

The left side $0.0053674$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.6.3

Compare the intervals to determine which ones satisfy the original inequality.

$x < 0$ True

$x > 0$ True

$x < 0$ True

$x > 0$ True

Step 2.7

The solution consists of all of the true intervals.

$x < 0$ or $x > 0$

Step 3

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 4