Calculus Examples

$f (x) = \ln (\ln (x))$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$\ln (x) = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

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

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

Step 1.2.2

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 1.2.3

Solve for $x$ .

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Step 1.2.3.1

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

$x = e^{0}$

Step 1.2.3.2

Anything raised to $0$ is $1$ .

$x = 1$

Step 1.3

The vertical asymptote occurs at $x = 1$ .

Vertical Asymptote: $x = 1$

Step 2

Find the point at $x = 2$ .

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Step 2.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = \ln (\ln (2))$

Step 2.2

The final answer is $\ln (\ln (2))$ .

$\ln (\ln (2))$

Step 2.3

Convert $\ln (\ln (2))$ to decimal.

$y = - 0.36651292$

Step 3

Find the point at $x = 3$ .

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Step 3.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \ln (\ln (3))$

Step 3.2

The final answer is $\ln (\ln (3))$ .

$\ln (\ln (3))$

Step 3.3

Convert $\ln (\ln (3))$ to decimal.

$y = 0.09404782$

Step 4

Find the point at $x = 4$ .

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Step 4.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = \ln (\ln (4))$

Step 4.2

The final answer is $\ln (\ln (4))$ .

$\ln (\ln (4))$

Step 4.3

Convert $\ln (\ln (4))$ to decimal.

$y = 0.32663425$

Step 5

The log function can be graphed using the vertical asymptote at $x = 1$ and the points $(2, - 0.36651292), (3, 0.09404782), (4, 0.32663425)$ .

Vertical Asymptote: $x = 1$

$\begin{array}{c} x & y \\ 2 & - 0.367 \\ 3 & 0.094 \\ 4 & 0.327 \end{array}$

Step 6