Problem 18 - College Algebra with Modeling and Visualization

Question

Find the domain of \(f\) and write it in setbuilder or interval notation. $$f(x)=\log _{5}\left(5^{x}-25\right)$$

Step-by-Step Solution

Answer

The domain of \( f(x) \) is \( (2, \infty) \).

Step 1: Understand the Function

The function given is a logarithmic function: \( f(x) = \log_{5}(5^x - 25) \). Logarithmic functions are defined only for positive arguments. This means \( 5^x - 25 > 0 \) must be true for \( f(x) \) to be defined.

Step 2: Solve the Inequality

We need to solve the inequality \( 5^x - 25 > 0 \). This simplifies to \( 5^x > 25 \). Recognizing that 25 can be rewritten as a power of 5, we have \( 25 = 5^2 \). Therefore, the inequality becomes \( 5^x > 5^2 \).

Step 3: Equate the Exponents

Since the bases (5) are the same, we can equate the exponents to solve \( x > 2 \). Thus, the solution to the inequality \( 5^x > 5^2 \) is \( x > 2 \).

Step 4: Determine the Domain

Given \( x > 2 \), the domain of the function \( f(x) \) is all real numbers greater than 2. In interval notation, the domain is represented as \( (2, \infty) \). In set-builder notation, it is \( \{ x \mid x > 2 \} \).

Key Concepts

Understanding Logarithmic Functions

A logarithmic function is a special type of function that involves the logarithm of a variable. The notation \( f(x) = \log_{b}(x) \) represents a logarithmic function, where \( b \) is the base of the logarithm and \( x \) is the variable. Logarithmic functions are the inverses of exponential functions.

In this context, we are dealing with \( f(x) = \log_{5}(5^x - 25) \). Before we can evaluate or work with this function, it must be defined for meaningful values of \( x \). A crucial property of logarithmic functions is that they are only defined for positive numbers. This means the expression inside the logarithm, \( 5^x - 25 \), must be greater than zero.

Therefore, our first step is ensuring that the argument of the logarithm is positive, which leads us to solve the inequality \( 5^x - 25 > 0 \). Understanding this requirement is key to working successfully with logarithmic functions.

Solving Inequalities Involving Exponents

Solving inequalities that involve exponential expressions can be a fun challenge. The inequality \( 5^x - 25 > 0 \) means we have to manipulate the expression so that it represents valid values for \( x \).

First, simplify the inequality: \( 5^x > 25 \). Now, we recognize that 25 can be rewritten using the same base as the exponential function, i.e., \( 25 = 5^2 \). Thus, the inequality becomes \( 5^x > 5^2 \).

When dealing with the same base in exponential form, we can compare the exponents directly. Therefore, this simplifies to \( x > 2 \). Converting the problem into terms of exponents allows us to solve it effectively and find the range of \( x \) that satisfies the given inequality.

Representing the Domain Using Interval Notation

Once we solve the inequality, \( x > 2 \), we can represent the domain of the function \( f(x) \) in various ways.

Interval notation is a concise way of representing a set of numbers, especially when dealing with continuous data. For the inequality \( x > 2 \), the domain is expressed in interval notation as \( (2, \infty) \). This notation indicates that \( x \) includes all real numbers greater than 2, extending indefinitely towards infinity.

Alternatively, set-builder notation explicitly describes the set of numbers that satisfies the condition. It would be \( \{ x \mid x > 2 \} \), which reads as "the set of all \( x \) such that \( x \) is greater than 2."

These forms of notation are powerful tools in mathematics, helping clearly convey the domain of a function and supporting better understanding among students.