Algebra Examples

f (x) = ln (x)

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

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

x = 0

$x = 0$

Step 1.2

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 (1)

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

Step 2.2

Simplify the result.

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Step 2.2.1

The natural logarithm of

1

$1$ is

0

$0$ .

f (1) = 0

$f (1) = 0$

Step 2.2.2

The final answer is

0

$0$ .

0

$0$

0

$0$

Step 2.3

Convert

0

$0$ to decimal.

y = 0

$y = 0$

y = 0

$y = 0$

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 (2)

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

Step 3.2

The final answer is

ln (2)

$\ln (2)$ .

ln (2)

$\ln (2)$

Step 3.3

Convert

ln (2)

$\ln (2)$ to decimal.

y = 0.69314718

$y = 0.69314718$

y = 0.69314718

$y = 0.69314718$

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)

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

Step 4.2

The final answer is

ln (3)

$\ln (3)$ .

ln (3)

$\ln (3)$

Step 4.3

Convert

ln (3)

$\ln (3)$ to decimal.

y = 1.09861228

$y = 1.09861228$

y = 1.09861228

$y = 1.09861228$

Step 5

The log function can be graphed using the vertical asymptote at

x = 0

$x = 0$ and the points

(1, 0), (2, 0.69314718), (3, 1.09861228)

$(1, 0), (2, 0.69314718), (3, 1.09861228)$ .

Vertical Asymptote:

x = 0

$x = 0$

\begin{matrix} x & y 1 & 0 2 & 0.693 3 & 1.099 \end{matrix}

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 0.693 \\ 3 & 1.099 \end{array}$

Step 6