Calculus Examples

$\ln (\frac{x}{3})$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$\frac{x}{3} = 0$

Step 1.2

Set the numerator equal to zero.

$x = 0$

Step 1.3

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Step 2

Find the point at $x = 3$ .

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Step 2.1

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

$f (3) = \ln (\frac{3}{3})$

Step 2.2

Simplify the result.

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Step 2.2.1

Divide $3$ by $3$ .

$f (3) = \ln (1)$

Step 2.2.2

The natural logarithm of $1$ is $0$ .

$f (3) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 1$ .

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Step 3.1

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

$f (1) = \ln (\frac{1}{3})$

Step 3.2

The final answer is $\ln (\frac{1}{3})$ .

$\ln (\frac{1}{3})$

Step 3.3

Convert $\ln (\frac{1}{3})$ to decimal.

$y = - 1.09861228$

Step 4

Find the point at $x = 2$ .

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Step 4.1

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

$f (2) = \ln (\frac{2}{3})$

Step 4.2

The final answer is $\ln (\frac{2}{3})$ .

$\ln (\frac{2}{3})$

Step 4.3

Convert $\ln (\frac{2}{3})$ to decimal.

$y = - 0.4054651$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(3, 0), (1, - 1.09861228), (2, - 0.4054651)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & - 1.099 \\ 2 & - 0.405 \\ 3 & 0 \end{array}$

Step 6