Calculus Examples

f (x) = ln (ln (x))

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

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

ln (x) = 0

$\ln (x) = 0$

Step 1.2

Solve for

x

$x$ .

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Step 1.2.1

To solve for

x

$x$ , rewrite the equation using properties of logarithms.

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

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

Step 1.2.2

Rewrite

ln (x) = 0

$\ln (x) = 0$ in exponential form using the definition of a logarithm. If

x

$x$ and

b

$b$ are positive real numbers and

b \neq 1

$b \neq 1$ , then

{log}_{b} (x) = y

$\log_{b} (x) = y$ is equivalent to

b^{y} = x

$b^{y} = x$ .

e^{0} = x

$e^{0} = x$

Step 1.2.3

Solve for

x

$x$ .

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Step 1.2.3.1

Rewrite the equation as

x = e^{0}

$x = e^{0}$ .

x = e^{0}

$x = e^{0}$

Step 1.2.3.2

Anything raised to

0

$0$ is

1

$1$ .

x = 1

$x = 1$

x = 1

$x = 1$

x = 1

$x = 1$

Step 1.3

The vertical asymptote occurs at

x = 1

$x = 1$ .

Vertical Asymptote:

x = 1

$x = 1$

Vertical Asymptote:

x = 1

$x = 1$

Step 2

Find the point at

x = 2

$x = 2$ .

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Step 2.1

Replace the variable

x

$x$ with

2

$2$ in the expression.

f (2) = ln (ln (2))

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

Step 2.2

The final answer is

ln (ln (2))

$\ln (\ln (2))$ .

ln (ln (2))

$\ln (\ln (2))$

Step 2.3

Convert

ln (ln (2))

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

y = - 0.36651292

$y = - 0.36651292$

y = - 0.36651292

$y = - 0.36651292$

Step 3

Find the point at

x = 3

$x = 3$ .

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Step 3.1

Replace the variable

x

$x$ with

3

$3$ in the expression.

f (3) = ln (ln (3))

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

Step 3.2

The final answer is

ln (ln (3))

$\ln (\ln (3))$ .

ln (ln (3))

$\ln (\ln (3))$

Step 3.3

Convert

ln (ln (3))

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

y = 0.09404782

$y = 0.09404782$

y = 0.09404782

$y = 0.09404782$

Step 4

Find the point at

x = 4

$x = 4$ .

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Step 4.1

Replace the variable

x

$x$ with

4

$4$ in the expression.

f (4) = ln (ln (4))

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

Step 4.2

The final answer is

ln (ln (4))

$\ln (\ln (4))$ .

ln (ln (4))

$\ln (\ln (4))$

Step 4.3

Convert

ln (ln (4))

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

y = 0.32663425

$y = 0.32663425$

y = 0.32663425

$y = 0.32663425$

Step 5

The log function can be graphed using the vertical asymptote at

x = 1

$x = 1$ and the points

(2, - 0.36651292), (3, 0.09404782), (4, 0.32663425)

$(2, - 0.36651292), (3, 0.09404782), (4, 0.32663425)$ .

Vertical Asymptote:

x = 1

$x = 1$

\begin{matrix} x & y 2 & - 0.367 3 & 0.094 4 & 0.327 \end{matrix}

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

Step 6