Problem 58 - Precalculus

Question

State the domain of the logarithmic function in interval notation. $$f(x)=\log _{3}(5-x)$$

Step-by-Step Solution

Answer

The domain of the function is \( (-\infty, 5) \).

Step 1: Understand the Logarithm Domain

The function \( f(x) = \log_{3}(5-x) \) is a logarithmic function of the form \( \log_{b}(a) \). For this function to be defined, the argument of the logarithm, \( a \), must be greater than zero. Therefore, we need to find for \( 5-x>0 \).

Step 2: Solve the Inequality

Solve the inequality \( 5-x > 0 \) for \( x \). Begin by isolating \( x \) on one side. Subtract 5 from both sides to get \( -x > -5 \). Then, multiply both sides by \( -1 \) and remember to flip the inequality sign, resulting in \( x < 5 \).

Step 3: Write the Domain in Interval Notation

Based on the inequality \( x < 5 \), the domain of \( f(x) = \log_{3}(5-x) \) consists of all real numbers less than 5. In interval notation, this is expressed as \( (-\infty, 5) \).

Key Concepts

Understanding Logarithmic Inequalities

In mathematics, a logarithmic inequality involves a logarithmic function, usually represented as \( \log_b(a) \), where \( b \) is the base and \( a \) is the argument of the logarithm. To solve these inequalities, one must ensure that the argument of the logarithm is positive because a logarithm is only defined for positive numbers. This is pivotal since the logarithmic function maps positive arguments to real numbers.

For example, consider the logarithmic function \( f(x) = \log_{3}(5-x) \). To find the domain of this function, you must solve the inequality \( 5-x > 0 \), indicating that \( 5-x \) must be greater than zero. This ensures that the argument of the logarithm is valid, hence \( x < 5 \) once solved, giving the permissible range for \( x \).

Understanding and solving logarithmic inequalities will help you determine the domain of logarithmic functions, setting the stage for accurate mathematical analysis.

Using Interval Notation

Once you solve the inequality, the next step is expressing the solution in interval notation. Interval notation is a compact and standardized way of describing a set of numbers. It is particularly useful in expressing domains and ranges in mathematics.

Here's how interval notation works:

• Brackets \([ \text{ ]} \text{ )} \) indicate whether endpoints are included in the set. Use square brackets \([\text{ ]}\) if the endpoint is included (closed interval), and use parentheses \((\text{ )})\) if it is not included (open interval).
• The infinity symbol \((\infty)\) always uses parentheses because infinity is a concept, not a number, and thus cannot be reached or included.

For the function \( f(x) = \log_{3}(5-x) \), after solving the inequality \( x < 5 \), the domain is expressed as \((-\infty, 5)\)—indicating all real numbers less than 5 but not including 5.

Function Domain Analysis

The domain of a function is the complete set of possible input values (\(x\)-values) that allow the function to work correctly. For logarithmic functions, this means that the argument must be positive, as logarithms of zero or negative numbers are undefined.

Let's walk through the domain analysis for the function \( f(x) = \log_{3}(5-x) \).

The main steps involved are:

• Identify the argument of the logarithmic function, which is \( 5-x \) in this case.
• Ensure it is positive by solving the inequality \( 5-x > 0 \).
• Isolate \( x \) by modifying the inequality, which results in \( x < 5 \) for this specific function.
• Express the domain in interval notation as \((-\infty, 5)\).

By conducting a thorough domain analysis, we ensure that the function remains valid and meaningful within its defined realm, providing a clearer understanding of its behavior and characteristics.