Algebra Examples

$\ln (x + 1)$

Find the asymptotes.

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Set the argument of the logarithm equal to zero.

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

The vertical asymptote occurs at $x = - 1$ .

Vertical Asymptote: $x = - 1$

Find the point at $x = 0$ .

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

$f (0) = \ln ((0) + 1)$

Simplify the result.

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Add $0$ and $1$ .

$f (0) = \ln (1)$

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

$f (0) = 0$

The final answer is $0$ .

$0$

Convert $0$ to decimal.

$y = 0$

Find the point at $x = 1$ .

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

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

Simplify the result.

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Add $1$ and $1$ .

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

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

$\ln (2)$

Convert $\ln (2)$ to decimal.

$y = 0.69314718$

Find the point at $x = 2$ .

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

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

Simplify the result.

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Add $2$ and $1$ .

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

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

$\ln (3)$

Convert $\ln (3)$ to decimal.

$y = 1.09861228$

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

Vertical Asymptote: $x = - 1$

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