Calculus Examples

\ln (3 x)

$\ln (3 x)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

3 x = 0

$3 x = 0$

Step 1.2

Divide each term in

3 x = 0

$3 x = 0$ by

3

$3$ and simplify.

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Step 1.2.1

Divide each term in

3 x = 0

$3 x = 0$ by

3

$3$ .

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

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of

3

$3$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

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

Step 1.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{0}{3}

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

x = \frac{0}{3}

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

x = \frac{0}{3}

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

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide

0

$0$ by

3

$3$ .

x = 0

$x = 0$

x = 0

$x = 0$

x = 0

$x = 0$

Step 1.3

The vertical asymptote occurs at

x = 0

$x = 0$ .

Vertical Asymptote:

x = 0

$x = 0$

Vertical Asymptote:

x = 0

$x = 0$

Step 2

Find the point at

x = 1

$x = 1$ .

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Step 2.1

Replace the variable

x

$x$ with

1

$1$ in the expression.

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply

3

$3$ by

1

$1$ .

f (1) = \ln (3)

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

Step 2.2.2

The final answer is

\ln (3)

$\ln (3)$ .

\ln (3)

$\ln (3)$

\ln (3)

$\ln (3)$

Step 2.3

Convert

\ln (3)

$\ln (3)$ to decimal.

y = 1.09861228

$y = 1.09861228$

y = 1.09861228

$y = 1.09861228$

Step 3

Find the point at

x = 2

$x = 2$ .

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Step 3.1

Replace the variable

x

$x$ with

2

$2$ in the expression.

f (2) = \ln (3 (2))

$f (2) = \ln (3 (2))$

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply

3

$3$ by

2

$2$ .

f (2) = \ln (6)

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

Step 3.2.2

The final answer is

\ln (6)

$\ln (6)$ .

\ln (6)

$\ln (6)$

\ln (6)

$\ln (6)$

Step 3.3

Convert

\ln (6)

$\ln (6)$ to decimal.

y = 1.79175946

$y = 1.79175946$

y = 1.79175946

$y = 1.79175946$

Step 4

Find the point at

x = 3

$x = 3$ .

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Step 4.1

Replace the variable

x

$x$ with

3

$3$ in the expression.

f (3) = \ln (3 (3))

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

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply

3

$3$ by

3

$3$ .

f (3) = \ln (9)

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

Step 4.2.2

The final answer is

\ln (9)

$\ln (9)$ .

\ln (9)

$\ln (9)$

\ln (9)

$\ln (9)$

Step 4.3

Convert

\ln (9)

$\ln (9)$ to decimal.

y = 2.19722457

$y = 2.19722457$

y = 2.19722457

$y = 2.19722457$

Step 5

The log function can be graphed using the vertical asymptote at

x = 0

$x = 0$ and the points

(1, 1.09861228), (2, 1.79175946), (3, 2.19722457)

$(1, 1.09861228), (2, 1.79175946), (3, 2.19722457)$ .

Vertical Asymptote:

x = 0

$x = 0$

\begin{array}{c} x & y \\ 1 & 1.099 \\ 2 & 1.792 \\ 3 & 2.197 \end{array}

$\begin{array}{c} x & y \\ 1 & 1.099 \\ 2 & 1.792 \\ 3 & 2.197 \end{array}$

Step 6