Precalculus Examples

$f (x) = \ln (x - 6)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x - 6 = 0$

Step 1.2

Add $6$ to both sides of the equation.

$x = 6$

Step 1.3

The vertical asymptote occurs at $x = 6$ .

Vertical Asymptote: $x = 6$

Step 2

Find the point at $x = 7$ .

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Step 2.1

Replace the variable $x$ with $7$ in the expression.

$f (7) = \ln ((7) - 6)$

Step 2.2

Simplify the result.

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Step 2.2.1

Subtract $6$ from $7$ .

$f (7) = \ln (1)$

Step 2.2.2

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

$f (7) = 0$

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 $x = 8$ .

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Step 3.1

Replace the variable $x$ with $8$ in the expression.

$f (8) = \ln ((8) - 6)$

Step 3.2

Simplify the result.

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Step 3.2.1

Subtract $6$ from $8$ .

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

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 $x = 9$ .

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Step 4.1

Replace the variable $x$ with $9$ in the expression.

$f (9) = \ln ((9) - 6)$

Step 4.2

Simplify the result.

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Step 4.2.1

Subtract $6$ from $9$ .

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

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 $x = 6$ and the points $(7, 0), (8, 0.69314718), (9, 1.09861228)$ .

Vertical Asymptote: $x = 6$

$\begin{array}{c} x & y \\ 7 & 0 \\ 8 & 0.693 \\ 9 & 1.099 \end{array}$

Step 6