Calculus Examples

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

Step 1

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

$\ln (\ln (x)) = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

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

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

Step 2.2

Rewrite $\ln (\ln (x)) = 0$ 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 $b^{y} = x$ .

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

Step 2.3

Solve for $x$ .

Tap for more steps...

Step 2.3.1

Rewrite the equation as $\ln (x) = e^{0}$ .

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

Step 2.3.2

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

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

Step 2.3.3

Rewrite $\ln (x) = e^{0}$ 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 $b^{y} = x$ .

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

Step 2.3.4

Solve for $x$ .

Tap for more steps...

Step 2.3.4.1

Rewrite the equation as $x = e^{e^{0}}$ .

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

Step 2.3.4.2

Simplify $e^{e^{0}}$ .

Tap for more steps...

Step 2.3.4.2.1

Anything raised to $0$ is $1$ .

$x = e^{1}$

Step 2.3.4.2.2

Simplify.

$x = e$

Step 3

Set the argument in $\ln (x)$ less than or equal to $0$ to find where the expression is undefined.

$x \leq 0$

Step 4

Set the argument in $\ln (\ln (x))$ less than or equal to $0$ to find where the expression is undefined.

$\ln (x) \leq 0$

Step 5

Solve for $x$ .

Tap for more steps...

Step 5.1

Convert the inequality to an equality.

$\ln (x) = 0$

Step 5.2

Solve the equation.

Tap for more steps...

Step 5.2.1

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

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

Step 5.2.2

Rewrite $\ln (x) = 0$ 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 $b^{y} = x$ .

$e^{0} = x$

Step 5.2.3

Solve for $x$ .

Tap for more steps...

Step 5.2.3.1

Rewrite the equation as $x = e^{0}$ .

$x = e^{0}$

Step 5.2.3.2

Anything raised to $0$ is $1$ .

$x = 1$

Step 5.3

Find the domain of $\ln (x)$ .

Tap for more steps...

Step 5.3.1

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

$x > 0$

Step 5.3.2

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

$(0, \infty)$

Step 5.4

Use each root to create test intervals.

$x < 0$

$0 < x < 1$

$x > 1$

Step 5.5

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 5.5.1

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

Tap for more steps...

Step 5.5.1.1

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

$x = - 2$

Step 5.5.1.2

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

$\ln (- 2) \leq 0$

Step 5.5.1.3

Determine if the inequality is true.

Tap for more steps...

Step 5.5.1.3.1

The equation cannot be solved because it is undefined.

$Undefined$

Step 5.5.1.3.2

The left side has no solution, which means that the given statement is false.

False

Step 5.5.2

Test a value on the interval $0 < x < 1$ to see if it makes the inequality true.

Tap for more steps...

Step 5.5.2.1

Choose a value on the interval $0 < x < 1$ and see if this value makes the original inequality true.

$x = 0.5$

Step 5.5.2.2

Replace $x$ with $0.5$ in the original inequality.

$\ln (0.5) \leq 0$

Step 5.5.2.3

The left side $- 0.69314718$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 5.5.3

Test a value on the interval $x > 1$ to see if it makes the inequality true.

Tap for more steps...

Step 5.5.3.1

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Step 5.5.3.2

Replace $x$ with $4$ in the original inequality.

$\ln (4) \leq 0$

Step 5.5.3.3

The left side $1.38629436$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 5.5.4

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

$x < 0$ False

$0 < x < 1$ True

$x > 1$ False

$x < 0$ False

$0 < x < 1$ True

$x > 1$ False

Step 5.6

The solution consists of all of the true intervals.

$0 < x \leq 1$

Step 6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 1, x = e$

$(- \infty, 1] \cup [e, e]$

Step 7