Problem 60 - Precalculus

Question

State the domain of the logarithmic function in interval notation. $$f(x)=\ln (3-x)$$

Step-by-Step Solution

Answer

The domain of the function is \((-\infty, 3)\).

Step 1: Identify the Condition for Logarithmic Functions

A logarithmic function \( f(x) = \ln (u) \) is defined if its argument \( u > 0 \). In this exercise, \( u = 3 - x \).

Step 2: Solve the Inequality for the Domain

Set up the inequality \( 3 - x > 0 \) to find the values of \( x \) for which the function is defined.

Step 3: Simplify the Inequality

To solve \( 3 - x > 0 \), first, isolate \( x \) by adding \( x \) to both sides to obtain \( 3 > x \). This can be rewritten as \( x < 3 \).

Step 4: Express the Domain in Interval Notation

The solution to \( x < 3 \) is expressed in interval notation as \( (-\infty, 3) \). This indicates that the function is defined for all \( x \) less than 3.

Key Concepts

Logarithmic Functions

Logarithmic functions are a fundamental aspect of mathematics, particularly in algebra and calculus. A logarithmic function is typically expressed in the form \( f(x) = \log_b(u) \), where \( b \) is the base, and \( u \) is a positive number. Most commonly, in more advanced applications, we use the natural logarithm, which has the base \( e \), and is written as \( \ln(u) \).

For the function to be valid or defined, the argument of the logarithm, which is the part inside the log, must be greater than zero. This is because logarithms are undefined for zero and negative numbers. Translating into our example, for \( f(x) = \ln(3-x) \), the expression inside the logarithm, \( 3-x \), must fulfill the condition \( 3-x > 0 \). Knowing these basic conditions helps us set up inequalities that reveal the domain of the function.

Inequalities in Mathematics

Inequalities are expressions used to define the relative size or order of two values or expressions. They are expressed using the symbols: \(<\), \(>\), \(\leq\), and \(\geq\). In the context of logarithmic functions, inequalities help determine where a function is defined.

Let's look at the inequality derived from \( f(x) = \ln(3-x) \). For this logarithmic function to be defined, we set up the inequality \( 3-x > 0 \). Solving inequalities involves manipulating the inequality using basic algebraic operations, while maintaining the truth of the statement.

• Add \( x \) to both sides to isolate \( x \): \( 3 > x \), which can also be read as \( x < 3 \).

Manipulating inequalities requires careful attention, as reversing the direction of the inequality is necessary when multiplying or dividing by negative numbers. Understanding inequalities lets us find the range of values, or domain, for which the logarithmic function is valid.

Interval Notation

Interval notation is a concise way of writing subsets of the real number line. It is frequently used to represent the domain and range of functions in mathematics. An interval is expressed using brackets or parentheses.

• Parentheses \((a, b)\) are used to indicate that the endpoints \(a\) and \(b\) are not included in the interval.
• Brackets \([a, b]\) indicate that the endpoints \(a\) and \(b\) are included.

For our function \( f(x) = \ln(3-x) \), the inequality solution gives \( x < 3 \). This is why the domain in interval notation is written as \( (-\infty, 3) \), denoting that all real numbers less than 3 are included, but 3 itself is not.

Interval notation simplifies understanding domain definitions and is a useful skill for anyone studying mathematics because it clearly lays out which values satisfy a given inequality.