Algebra Examples

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

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x + 2 = 0$

Step 1.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.3

The vertical asymptote occurs at $x = - 2$ .

Vertical Asymptote: $x = - 2$

Step 2

Find the point at $x = - 1$ .

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Step 2.1

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

$f (- 1) = \ln ((- 1) + 2)$

Step 2.2

Simplify the result.

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Step 2.2.1

Add $- 1$ and $2$ .

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

Step 2.2.2

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

$f (- 1) = 0$

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 $x = 0$ .

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Step 3.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \ln ((0) + 2)$

Step 3.2

Simplify the result.

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Step 3.2.1

Add $0$ and $2$ .

$f (0) = \ln (2)$

Step 3.2.2

The final answer is $\ln (2)$ .

$\ln (2)$

Step 3.3

Convert $\ln (2)$ to decimal.

$y = 0.69314718$

Step 4

Find the point at $x = 1$ .

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Step 4.1

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

$f (1) = \ln ((1) + 2)$

Step 4.2

Simplify the result.

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Step 4.2.1

Add $1$ and $2$ .

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

Step 4.2.2

The final answer is $\ln (3)$ .

$\ln (3)$

Step 4.3

Convert $\ln (3)$ to decimal.

$y = 1.09861228$

Step 5

The log function can be graphed using the vertical asymptote at $x = - 2$ and the points $(- 1, 0), (0, 0.69314718), (1, 1.09861228)$ .

Vertical Asymptote: $x = - 2$

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

Step 6