Calculus Examples

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

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

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

Step 1.2

Solve for $x$ .

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Step 1.2.1

Write in exponential form.

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Step 1.2.1.1

For logarithmic equations, $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ such that $x > 0$ , $b > 0$ , and $b \neq 1$ . In this case, $b = e$ , $x = x^{2}$ , and $y = 0$ .

$b = e$

$x = x^{2}$

$y = 0$

Step 1.2.1.2

Substitute the values of $b$ , $x$ , and $y$ into the equation $b^{y} = x$ .

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

Step 1.2.2

Solve for $x$ .

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Step 1.2.2.1

Rewrite the equation as $x^{2} = e^{0}$ .

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

Step 1.2.2.2

Anything raised to $0$ is $1$ .

$x^{2} = 1$

Step 1.2.2.3

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

$x = \pm \sqrt{1}$

Step 1.2.2.4

Any root of $1$ is $1$ .

$x = \pm 1$

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 $\pm$ to find the first solution.

$x = 1$

Step 1.2.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

Step 1.2.2.5.3

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

$x = 1, - 1$

Step 1.3

The vertical asymptote occurs at $x = 1, x = - 1$ .

Vertical Asymptote: $x = 1, x = - 1$

Step 2

Find the point at $x = - 2$ .

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Step 2.1

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Raise $- 2$ to the power of $2$ .

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

Step 2.2.2

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

$\ln (\ln (4))$

Step 2.3

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

$= 0.32663425$

Step 3

Find the point at $x = - 3$ .

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

Raise $- 3$ to the power of $2$ .

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

Step 3.2.2

The final answer is $\ln (\ln (9))$ .

$\ln (\ln (9))$

Step 3.3

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

$= 0.787195$

Step 4

Find the point at $x = - 4$ .

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Raise $- 4$ to the power of $2$ .

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

Step 4.2.2

The final answer is $\ln (\ln (16))$ .

$\ln (\ln (16))$

Step 4.3

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

$= 1.01978144$

Step 5

The log function can be graphed using the vertical asymptote at $x = 1, x = - 1$ and the points $(- 2, 0.32663425), (- 3, 0.787195), (- 4, 1.01978144)$ .

Vertical Asymptote: $x = 1, x = - 1$

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

Step 6