Problem 14 - College Algebra with Modeling and Visualization

Question

Find the domain of \(f\) and write it in setbuilder or interval notation. $$f(x)=\ln (2 x-4)$$

Step-by-Step Solution

Answer

The domain is \( (2, \infty) \) or \( \{ x \mid x > 2 \} \).

Step 1: Understand the Domain of a Logarithm Function

For the logarithmic function \(\ln(x)\), the expression inside the natural logarithm (i.e., \(x\)) must be greater than zero. This ensures that the logarithm function is defined and real-valued. Therefore, for \(f(x) = \ln(2x - 4)\), we need the argument \((2x - 4)\) to satisfy \(2x - 4 > 0\).

Step 2: Set Up the Inequality

To find the domain, set up the inequality for the logarithmic function so that the expression inside is greater than zero: \(2x - 4 > 0\). This inequality represents the condition under which the logarithm is defined.

Step 3: Solve the Inequality

Solve the inequality \(2x - 4 > 0\) to find the values of \(x\) that satisfy this condition. Add 4 to both sides to get \(2x > 4\). Then, divide both sides by 2 to isolate \(x\), resulting in \(x > 2\).

Step 4: Express the Domain in Interval Notation

Based on the inequality \(x > 2\), the domain of the function can be expressed in interval notation. Since \(x\) can take any value greater than 2, the domain is \((2, \infty)\).

Step 5: Express the Domain in Set Builder Notation

Alternatively, express the domain in set builder notation for completeness: \(\{ x \mid x > 2 \}\). This notation describes the set of all \(x\) such that \(x\) is greater than 2.

Key Concepts

Logarithmic Function

A logarithmic function is a mathematical function that involves the logarithm of a variable. Logarithms tell us how many times we multiply a base number to achieve a specific value. The function is typically represented as \( \log_b(x) \), where \( b \) is the base and \( x \) is the argument of the logarithm. For natural logarithms, the base \( e \), which is approximately 2.718, is used, and these are denoted by \( \ln(x) \).

In general, the domain of a logarithmic function like \( \ln(x) \) is restricted because logarithms can only be taken of positive values in elementary mathematics. This means that the argument \( x \) must be greater than zero. If we look at \( f(x) = \ln(2x - 4) \), we need the expression \( 2x - 4 \) within the logarithm to be positively valued. From understanding this restriction, it is clear why this step forms the basis for determining the function's domain.

Inequality Solving

Solving inequalities is a crucial skill in mathematics that allows us to determine the range of values that satisfy a particular condition. In our example, solving the inequality helps us find where the logarithmic expression is positive.

Starting with the inequality \( 2x - 4 > 0 \), we perform algebraic operations to isolate \( x \).

• Add 4 to both sides: \( 2x > 4 \).
• Divide both sides by 2: \( x > 2 \).

This process shows that \( x \) must be greater than 2 for the logarithmic function to be real and defined. Mastering the art of solving inequalities is fundamental, as it establishes essential conditions for mathematical expressions and functions.

Interval Notation

Interval notation is a way of representing the set of values that a variable can assume. In this notation, we use parentheses \(( )\) and square brackets \([ ]\) to express intervals.

For the domain of the logarithmic function \( f(x) = \ln(2x - 4) \), we found that \( x > 2 \). In interval notation, this is written as \((2, \infty)\).

• Parentheses \(( )\) indicate that the endpoints are not included. Thus, \( 2 \) is not part of the domain because the function is undefined at \( x = 2 \).
• The symbol \( \infty \) represents infinity, indicating that there is no upper boundary other than positive infinity.

Interval notation provides a concise and clear way to express the domain of functions, especially useful when dealing with real numbers.

Set-Builder Notation

Set-builder notation is another method used to define sets where a set is described by a rule or a property that its elements must satisfy. It elegantly describes the collection of elements that meet certain criteria.

In the case of our logarithmic function, once we determined that \( x > 2 \), we expressed the domain in set-builder notation as \( \{ x \mid x > 2 \} \). Here, the vertical bar \( \mid \) can be read as "such that." This notation indicates "the set of all numbers \( x \) such that \( x \) is greater than 2."

Set-builder notation is particularly useful in pure mathematics because it precisely defines sets with various properties, offering clarity when intricate conditions are involved.