Trigonometry Examples

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

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x^{3} = 0$

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.

$x = \sqrt[3]{0}$

Step 1.2.2

Simplify

$\sqrt[3]{0}$ .

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Step 1.2.2.1

Rewrite

$0$ as

$0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 1.2.2.2

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

$x = 0$

Step 1.3

The vertical asymptote occurs at

$x = 0$ .

Vertical Asymptote:

$x = 0$

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.

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

Step 2.2

Simplify the result.

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Step 2.2.1

The natural logarithm of

$1$ is

$0$ .

$f (1) = 3 \cdot 0$

Step 2.2.2

Multiply

$3$ by

$0$ .

$f (1) = 0$

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

$x = 2$ .

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Step 3.1

Replace the variable

$x$ with

$2$ in the expression.

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

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify

$3 \ln (2)$ by moving

$3$ inside the logarithm.

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

Step 3.2.2

Raise

$2$ to the power of

$3$ .

$f (2) = \ln (8)$

Step 3.2.3

The final answer is

$\ln (8)$ .

$\ln (8)$

Step 3.3

Convert

$\ln (8)$ to decimal.

$y = 2.07944154$

Step 4

Find the point at

$x = 3$ .

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Step 4.1

Replace the variable

$x$ with

$3$ in the expression.

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

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify

$3 \ln (3)$ by moving

$3$ inside the logarithm.

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

Step 4.2.2

Raise

$3$ to the power of

$3$ .

$f (3) = \ln (27)$

Step 4.2.3

The final answer is

$\ln (27)$ .

$\ln (27)$

Step 4.3

Convert

$\ln (27)$ to decimal.

$y = 3.29583686$

Step 5

The log function can be graphed using the vertical asymptote at

$x = 0$ and the points

$(1, 0), (2, 2.07944154), (3, 3.29583686)$ .

Vertical Asymptote:

$x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 2.079 \\ 3 & 3.296 \end{array}$

Step 6