Precalculus Examples

$g (x) = \log_{5} (x - 4) - 3$

Step 1

Find the asymptotes.

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Step 1.1

Find where the expression $\log_{5} (x - 4) - 3$ is undefined.

$x \leq 4$

Step 1.2

Since $\log_{5} (x - 4) - 3$ $\to$ $\infty$ as $x$ $\to$ $4$ from the left and $\log_{5} (x - 4) - 3$ $\to$ $- \infty$ as $x$ $\to$ $4$ from the right, then $x = 4$ is a vertical asymptote.

$x = 4$

Step 1.3

Ignoring the logarithm, consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 1.4

There are no horizontal asymptotes because $Q (x)$ is $1$ .

No Horizontal Asymptotes

Step 1.5

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

Step 1.6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 4$

No Horizontal Asymptotes

Vertical Asymptotes: $x = 4$

No Horizontal Asymptotes

Step 2

Find the point at $x = 5$ .

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Step 2.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = \log_{5} ((5) - 4) - 3$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Subtract $4$ from $5$ .

$f (5) = \log_{5} (1) - 3$

Step 2.2.1.2

Logarithm base $5$ of $1$ is $0$ .

$f (5) = 0 - 3$

Step 2.2.2

Subtract $3$ from $0$ .

$f (5) = - 3$

Step 2.2.3

The final answer is $- 3$ .

$- 3$

Step 2.3

Convert $- 3$ to decimal.

$y = - 3$

Step 3

Find the point at $x = 9$ .

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Step 3.1

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

$f (9) = \log_{5} ((9) - 4) - 3$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Subtract $4$ from $9$ .

$f (9) = \log_{5} (5) - 3$

Step 3.2.1.2

Logarithm base $5$ of $5$ is $1$ .

$f (9) = 1 - 3$

Step 3.2.2

Subtract $3$ from $1$ .

$f (9) = - 2$

Step 3.2.3

The final answer is $- 2$ .

$- 2$

Step 3.3

Convert $- 2$ to decimal.

$y = - 2$

Step 4

Find the point at $x = 6$ .

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Step 4.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = \log_{5} ((6) - 4) - 3$

Step 4.2

Simplify the result.

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Step 4.2.1

Subtract $4$ from $6$ .

$f (6) = \log_{5} (2) - 3$

Step 4.2.2

The final answer is $\log_{5} (2) - 3$ .

$\log_{5} (2) - 3$

Step 4.3

Convert $\log_{5} (2) - 3$ to decimal.

$y = - 2.56932344$

Step 5

The log function can be graphed using the vertical asymptote at $x = 4$ and the points $(5, - 3), (9, - 2), (6, - 2.56932344)$ .

Vertical Asymptote: $x = 4$

$\begin{array}{c} x & y \\ 5 & - 3 \\ 6 & - 2.569 \\ 9 & - 2 \end{array}$

Step 6