Calculus Examples

ln (ln (x^{2}))

$\ln (\ln (x^{2}))$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

ln (x^{2}) = 0

$\ln (x^{2}) = 0$

Step 1.2

Solve for

x

$x$ .

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Step 1.2.1

Write in exponential form.

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Step 1.2.1.1

For logarithmic equations,

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

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

b^{y} = x

$b^{y} = x$ such that

x > 0

$x > 0$ ,

b > 0

$b > 0$ , and

b \neq 1

$b \neq 1$ . In this case,

b = e

$b = e$ ,

x = x^{2}

$x = x^{2}$ , and

y = 0

$y = 0$ .

b = e

$b = e$

x = x^{2}

$x = x^{2}$

y = 0

$y = 0$

Step 1.2.1.2

Substitute the values of

b

$b$ ,

x

$x$ , and

y

$y$ into the equation

b^{y} = x

$b^{y} = x$ .

e^{0} = x^{2}

$e^{0} = x^{2}$

e^{0} = x^{2}

$e^{0} = x^{2}$

Step 1.2.2

Solve for

x

$x$ .

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Step 1.2.2.1

Rewrite the equation as

x^{2} = e^{0}

$x^{2} = e^{0}$ .

x^{2} = e^{0}

$x^{2} = e^{0}$

Step 1.2.2.2

Anything raised to

0

$0$ is

1

$1$ .

x^{2} = 1

$x^{2} = 1$

Step 1.2.2.3

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

x = \pm \sqrt{1}

$x = \pm \sqrt{1}$

Step 1.2.2.4

Any root of

1

$1$ is

1

$1$ .

x = \pm 1

$x = \pm 1$

Step 1.2.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.2.5.1

First, use the positive value of the

\pm

$\pm$ to find the first solution.

x = 1

$x = 1$

Step 1.2.2.5.2

Next, use the negative value of the

\pm

$\pm$ to find the second solution.

x = - 1

$x = - 1$

Step 1.2.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

x = 1, - 1

$x = 1, - 1$

x = 1, - 1

$x = 1, - 1$

x = 1, - 1

$x = 1, - 1$

x = 1, - 1

$x = 1, - 1$

Step 1.3

The vertical asymptote occurs at

x = 1, x = - 1

$x = 1, x = - 1$ .

Vertical Asymptote:

x = 1, x = - 1

$x = 1, x = - 1$

Vertical Asymptote:

x = 1, x = - 1

$x = 1, x = - 1$

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)}^{2}))

$f (- 2) = \ln (\ln ({(- 2)}^{2}))$

Step 2.2

Simplify the result.

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Step 2.2.1

Raise

- 2

$- 2$ to the power of

2

$2$ .

f (- 2) = ln (ln (4))

$f (- 2) = \ln (\ln (4))$

Step 2.2.2

The final answer is

ln (ln (4))

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

ln (ln (4))

$\ln (\ln (4))$

ln (ln (4))

$\ln (\ln (4))$

Step 2.3

Convert

ln (ln (4))

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

= 0.32663425

$= 0.32663425$

= 0.32663425

$= 0.32663425$

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)}^{2}))

$f (- 3) = \ln (\ln ({(- 3)}^{2}))$

Step 3.2

Simplify the result.

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Step 3.2.1

Raise

- 3

$- 3$ to the power of

2

$2$ .

f (- 3) = ln (ln (9))

$f (- 3) = \ln (\ln (9))$

Step 3.2.2

The final answer is

ln (ln (9))

$\ln (\ln (9))$ .

ln (ln (9))

$\ln (\ln (9))$

ln (ln (9))

$\ln (\ln (9))$

Step 3.3

Convert

ln (ln (9))

$\ln (\ln (9))$ to decimal.

= 0.787195

$= 0.787195$

= 0.787195

$= 0.787195$

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)}^{2}))

$f (- 4) = \ln (\ln ({(- 4)}^{2}))$

Step 4.2

Simplify the result.

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Step 4.2.1

Raise

- 4

$- 4$ to the power of

2

$2$ .

f (- 4) = ln (ln (16))

$f (- 4) = \ln (\ln (16))$

Step 4.2.2

The final answer is

ln (ln (16))

$\ln (\ln (16))$ .

ln (ln (16))

$\ln (\ln (16))$

ln (ln (16))

$\ln (\ln (16))$

Step 4.3

Convert

ln (ln (16))

$\ln (\ln (16))$ to decimal.

= 1.01978144

$= 1.01978144$

= 1.01978144

$= 1.01978144$

Step 5

The log function can be graphed using the vertical asymptote at

x = 1, x = - 1

$x = 1, x = - 1$ and the points

(- 2, 0.32663425), (- 3, 0.787195), (- 4, 1.01978144)

$(- 2, 0.32663425), (- 3, 0.787195), (- 4, 1.01978144)$ .

Vertical Asymptote:

x = 1, x = - 1

$x = 1, x = - 1$

\begin{matrix} x & y - 4 & 1.02 - 3 & 0.787 - 2 & 0.327 \end{matrix}

$\begin{array}{c} x & y \\ - 4 & 1.02 \\ - 3 & 0.787 \\ - 2 & 0.327 \end{array}$

Step 6