Precalculus Examples

$\ln (\ln (x))$

Step 1

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

$x > 0$

Step 2

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

$\ln (x) > 0$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Convert the inequality to an equality.

$\ln (x) = 0$

Step 3.2

Solve the equation.

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

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 3.2.3

Solve for $x$ .

Tap for more steps...

Step 3.2.3.1

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

$x = e^{0}$

Step 3.2.3.2

Anything raised to $0$ is $1$ .

$x = 1$

Step 3.3

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

Tap for more steps...

Step 3.3.1

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

$x > 0$

Step 3.3.2

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

$(0, \infty)$

Step 3.4

The solution consists of all of the true intervals.

$x > 1$

Step 4

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

Interval Notation:

$(1, \infty)$

Set-Builder Notation:

${x | x > 1}$

Step 5