Calculus Examples

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

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x^{2} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

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

$x = \pm \sqrt{0}$

Step 1.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.2.2

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

$x = \pm 0$

Step 1.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.3

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Step 2

Find the point at $x = 1$ .

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Step 2.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \ln ({(t)}^{2})$

Step 2.2

Simplify the result.

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Step 2.2.1

Remove parentheses.

$f (1) = \ln (t^{2})$

Step 2.2.2

The final answer is $\ln (t^{2})$ .

$\ln (t^{2})$

Step 2.3

Convert $\ln (t^{2})$ to decimal.

$y = \ln (t^{2})$

Step 3

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, \ln (t^{2})), (2, \ln (t^{2})), (3, \ln (t^{2}))$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & \ln (t^{2}) \\ 2 & \ln (t^{2}) \\ 3 & \ln (t^{2}) \end{array}$

Step 4