Calculus Examples

$\ln (6 x)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$6 x = 0$

Step 1.2

Divide each term in $6 x = 0$ by $6$ and simplify.

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Step 1.2.1

Divide each term in $6 x = 0$ by $6$ .

$\frac{6 x}{6} = \frac{0}{6}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $6$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{0}{6}$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{6}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $0$ by $6$ .

$x = 0$

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.

$f (1) = \ln (6 (1))$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $6$ by $1$ .

$f (1) = \ln (6)$

Step 2.2.2

The final answer is $\ln (6)$ .

$\ln (6)$

Step 2.3

Convert $\ln (6)$ to decimal.

$y = 1.79175946$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = \ln (6 (2))$

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $6$ by $2$ .

$f (2) = \ln (12)$

Step 3.2.2

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

$\ln (12)$

Step 3.3

Convert $\ln (12)$ to decimal.

$y = 2.48490664$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = \ln (6 (3))$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $6$ by $3$ .

$f (3) = \ln (18)$

Step 4.2.2

The final answer is $\ln (18)$ .

$\ln (18)$

Step 4.3

Convert $\ln (18)$ to decimal.

$y = 2.89037175$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 1.79175946), (2, 2.48490664), (3, 2.89037175)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 1.792 \\ 2 & 2.485 \\ 3 & 2.89 \end{array}$

Step 6