Calculus Examples

$\ln (y)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$y = 0$

Step 1.2

The vertical asymptote occurs at $y = 0$ .

Vertical Asymptote: $y = 0$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

$f (1) = e^{1}$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify.

$f (1) = e$

Step 2.2.2

The final answer is $e$ .

$e$

Step 2.3

Convert $e$ to decimal.

$y = 2.71828182$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = e^{2}$

Step 3.2

The final answer is $e^{2}$ .

$e^{2}$

Step 3.3

Convert $e^{2}$ to decimal.

$y = 7.38905609$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = e^{3}$

Step 4.2

The final answer is $e^{3}$ .

$e^{3}$

Step 4.3

Convert $e^{3}$ to decimal.

$y = 20.08553692$

Step 5

The log function can be graphed using the vertical asymptote at $y = 0$ and the points $(1, 2.71828182), (2, 7.38905609), (3, 20.08553692)$ .

Vertical Asymptote: $y = 0$

$\begin{array}{c} x & y \\ 1 & 2.718 \\ 2 & 7.389 \\ 3 & 20.086 \end{array}$

Step 6