Calculus Examples

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

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x^{2} - 4 = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Add $4$ to both sides of the equation.

$x^{2} = 4$

Step 1.2.2

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

$x = \pm \sqrt{4}$

Step 1.2.3

Simplify $\pm \sqrt{4}$ .

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Step 1.2.3.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2}}$

Step 1.2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 1.2.4

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

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Step 1.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 1.2.4.2

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

$x = - 2$

Step 1.2.4.3

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

$x = 2, - 2$

Step 1.3

The vertical asymptote occurs at $x = 2, x = - 2$ .

Vertical Asymptote: $x = 2, x = - 2$

Step 2

Find the point at $x = - 3$ .

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Step 2.1

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

$f (- 3) = \ln ({(- 3)}^{2} - 4)$

Step 2.2

Simplify the result.

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Step 2.2.1

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

$f (- 3) = \ln (9 - 4)$

Step 2.2.2

Subtract $4$ from $9$ .

$f (- 3) = \ln (5)$

Step 2.2.3

The final answer is $\ln (5)$ .

$\ln (5)$

Step 2.3

Convert $\ln (5)$ to decimal.

$= 1.60943791$

Step 3

Find the point at $x = - 4$ .

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

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

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

Step 3.2.2

Subtract $4$ from $16$ .

$f (- 4) = \ln (12)$

Step 3.2.3

The final answer is $\ln (12)$ .

$\ln (12)$

Step 3.3

Convert $\ln (12)$ to decimal.

$= 2.48490664$

Step 4

Find the point at $x = - 5$ .

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Step 4.1

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = \ln ({(- 5)}^{2} - 4)$

Step 4.2

Simplify the result.

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Step 4.2.1

Raise $- 5$ to the power of $2$ .

$f (- 5) = \ln (25 - 4)$

Step 4.2.2

Subtract $4$ from $25$ .

$f (- 5) = \ln (21)$

Step 4.2.3

The final answer is $\ln (21)$ .

$\ln (21)$

Step 4.3

Convert $\ln (21)$ to decimal.

$= 3.04452243$

Step 5

The log function can be graphed using the vertical asymptote at $x = 2, x = - 2$ and the points $(- 3, 1.60943791), (- 4, 2.48490664), (- 5, 3.04452243)$ .

Vertical Asymptote: $x = 2, x = - 2$

$\begin{array}{c} x & y \\ - 5 & 3.045 \\ - 4 & 2.485 \\ - 3 & 1.609 \end{array}$

Step 6