Precalculus Examples

$f (x) = - 2 \log_{\frac{1}{3}} (x - 1) - 1$

Step 1

Find the asymptotes.

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Step 1.1

Find where the expression $- \log_{\frac{1}{3}} ({(x - 1)}^{2}) - 1$ is undefined.

$x = 1$

Step 1.2

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 1.3

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

No Horizontal Asymptotes

Step 1.4

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

Step 1.5

This is the set of all asymptotes.

Vertical Asymptotes: $x = 1$

No Horizontal Asymptotes

Vertical Asymptotes: $x = 1$

No Horizontal Asymptotes

Step 2

Find the point at $x = 2$ .

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Step 2.1

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

$f (2) = - 2 \log_{\frac{1}{3}} ((2) - 1) - 1$

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 $1$ from $2$ .

$f (2) = - 2 \log_{\frac{1}{3}} (1) - 1$

Step 2.2.1.2

Logarithm base $\frac{1}{3}$ of $1$ is $0$ .

$f (2) = - 2 \cdot 0 - 1$

Step 2.2.1.3

Multiply $- 2$ by $0$ .

$f (2) = 0 - 1$

Step 2.2.2

Subtract $1$ from $0$ .

$f (2) = - 1$

Step 2.2.3

The final answer is $- 1$ .

$- 1$

Step 2.3

Convert $- 1$ to decimal.

$y = - 1$

Step 3

Find the point at $x = 4$ .

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Step 3.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = - 2 \log_{\frac{1}{3}} ((4) - 1) - 1$

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 $1$ from $4$ .

$f (4) = - 2 \log_{\frac{1}{3}} (3) - 1$

Step 3.2.1.2

Logarithm base $\frac{1}{3}$ of $3$ is $- 1$ .

$f (4) = - 2 \cdot - 1 - 1$

Step 3.2.1.3

Multiply $- 2$ by $- 1$ .

$f (4) = 2 - 1$

Step 3.2.2

Subtract $1$ from $2$ .

$f (4) = 1$

Step 3.2.3

The final answer is $1$ .

$1$

Step 3.3

Convert $1$ to decimal.

$y = 1$

Step 4

Find the point at $x = 10$ .

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Step 4.1

Replace the variable $x$ with $10$ in the expression.

$f (10) = - 2 \log_{\frac{1}{3}} ((10) - 1) - 1$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Subtract $1$ from $10$ .

$f (10) = - 2 \log_{\frac{1}{3}} (9) - 1$

Step 4.2.1.2

Logarithm base $\frac{1}{3}$ of $9$ is $- 2$ .

$f (10) = - 2 \cdot - 2 - 1$

Step 4.2.1.3

Multiply $- 2$ by $- 2$ .

$f (10) = 4 - 1$

Step 4.2.2

Subtract $1$ from $4$ .

$f (10) = 3$

Step 4.2.3

The final answer is $3$ .

$3$

Step 4.3

Convert $3$ to decimal.

$y = 3$

Step 5

The log function can be graphed using the vertical asymptote at $x = 1$ and the points $(2, - 1), (4, 1), (10, 3)$ .

Vertical Asymptote: $x = 1$

$\begin{array}{c} x & y \\ 2 & - 1 \\ 4 & 1 \\ 10 & 3 \end{array}$

Step 6