Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$f(x)=\log |3 x-7|$$

Step-by-Step Solution

Answer

The domain is \( x \in (-\infty, \frac{7}{3}) \cup (\frac{7}{3}, \infty) \).

Step 1: Understand the Function

The function given is \( f(x) = \log |3x - 7| \). This is a logarithmic function involving the absolute value of the linear expression \(3x - 7\).

Step 2: Determine the Conditions for the Logarithm

For a logarithmic function \( \log b \) to be defined, the argument \( b \) must be strictly positive. Therefore, for \( \log |3x - 7| \) to be defined, \( |3x - 7| > 0 \) must hold. Since the absolute value of any non-zero expression is always positive, we need \( 3x - 7 eq 0 \).

Step 3: Solve the Inequality

The inequality \( |3x - 7| > 0 \) is satisfied when \( 3x - 7 eq 0 \). Solving this, we find \( 3x eq 7 \), leading to \( x eq \frac{7}{3} \). Thus, \( x \) can be any real number except \( \frac{7}{3} \).

Step 4: Express the Domain

The domain of the function is the set of all real numbers except \( x = \frac{7}{3} \). Therefore, the domain can be written as \( x \in (-\infty, \frac{7}{3}) \cup (\frac{7}{3}, \infty) \).

Step 5: (Optional) Graphical Representation

Graphing the function \( f(x) = \log |3x - 7| \) confirms that there is a vertical asymptote at \( x = \frac{7}{3} \). The graph exists on both sides of this asymptote, supporting our analytical solution of the domain.

Key Concepts

Logarithmic Functions

Logarithmic functions are fascinating mathematical expressions that involve the logarithm, typically expressed as \( \log_{b}(x) \). In these functions, \( b \) represents the base, and \( x \) is the argument or input value. Logarithms are essentially the inverse of exponentiation, helping to solve equations where the unknown variable is an exponent.
An interesting aspect of logarithmic functions is their domain. Since the logarithm is only defined for positive arguments,

• the input value \( x \) must be greater than zero.
• This condition emerges naturally from the definition of a logarithm.

In essence, a logarithmic function restricts its domain to positive numbers.

Absolute Value

The concept of absolute value is a key component in the function \( f(x) = \log |3x - 7| \). Absolute value denotes the non-negative magnitude of a number, essentially ignoring its sign. For instance, both \( |-5| \) and \( |5| \) equal 5.
In the context of our function, the expression \( |3x - 7| \) mandates that whatever value emerges after computations, it's considered positive.
While designing the domain, you must focus on cases where the expression under the absolute value doesn't equal zero, ensuring the logarithm remains defined.

• This introduces conditions such as \( 3x - 7 eq 0 \).
• The absolute value ensures that once outside the range of zero, values remain strictly positive, satisfying the logarithmic requirement.

Graphical Representation

Visualizing the function helps reinforce understanding. Graphical representation translates mathematical solutions into visual ones, providing additional insights. When you graph \( f(x) = \log |3x - 7| \), you will notice a vertical asymptote at \( x = \frac{7}{3} \).
Asymptotes are lines that the graph approaches but never actually touches. This specific asymptote reflects the point where \( 3x - 7 = 0 \), making \( x = \frac{7}{3} \) a critical point where the function isn't defined.
Seeing both halves of the graph, you can observe the behavior of \( f(x) \) as \( x \) approaches and then diverts from the asymptote. This graphical view confirms the analytical result of the domain as all real numbers except \( x = \frac{7}{3} \).

Inequality Solving

Solving inequalities is essential for determining domains when functions include absolute values or conditions like \( |3x - 7| > 0 \). By setting the condition from the logarithmic argument, you create an inequality that must be satisfied.
The inequality \( |3x - 7| > 0 \) implies \( 3x - 7 eq 0 \), as any non-zero value taken inside an absolute value ensures the result is positive.
To solve:

• Transform the inequality into \( 3x - 7 eq 0 \).
• Rearrange to find \( 3x eq 7 \).
• Finally, solve: \( x eq \frac{7}{3} \).

This way, you can express the domain as all real numbers excluding \( \frac{7}{3} \), supporting the function's defined nature.