Finding The Domain Of F(x) = Log₃(2x-3): A Step-by-Step Guide

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Finding the Domain of f(x) = log₃(2x-3): A Step-by-Step Guide

Hey guys! Today, we're diving into a common yet crucial topic in mathematics: finding the domain of a logarithmic function. Specifically, we'll be tackling the function f(x) = log₃(2x-3). Understanding how to determine the domain is super important because it tells us for which values of 'x' our function actually gives us a real output. So, let's get started and break this down step by step!

Understanding the Basics of Logarithmic Functions

Before we jump into solving our specific problem, let's quickly recap what logarithmic functions are all about. Logarithmic functions are, in essence, the inverse of exponential functions. Think of it like this: if exponentiation is like raising a number to a power, logarithms are like figuring out what power you need to raise the base to in order to get a certain number. The general form of a logarithmic function is:

f(x) = logₐ(x)

Where:

  • 'a' is the base of the logarithm (a > 0 and a ≠ 1).
  • 'x' is the argument or the value we're taking the logarithm of.

The most critical thing to remember about logarithms is that they are only defined for positive arguments. You can't take the logarithm of a negative number or zero. This is because there's no power you can raise a positive base to that will give you a non-positive result. This fundamental rule is the key to finding the domain of our function.

Logarithmic functions are used extensively in various fields, including computer science, engineering, and finance. They're particularly useful for dealing with quantities that vary over a large range, as logarithms can compress these scales and make them easier to work with. Furthermore, logarithms play a crucial role in solving exponential equations and modeling real-world phenomena such as compound interest, radioactive decay, and the Richter scale for measuring earthquake intensity. Understanding logarithmic functions not only helps in advanced mathematical studies but also provides tools for tackling practical problems across different domains. So, grasping the concept of domains for logarithmic functions is more than just an academic exercise; it's a valuable skill that can be applied in many real-world scenarios. Keep in mind that the base of the logarithm significantly affects the function's behavior and domain considerations, reinforcing the importance of a solid understanding of the underlying principles of logarithmic functions.

Identifying the Key Restriction for f(x) = log₃(2x-3)

Okay, now that we've refreshed our knowledge of logarithmic functions, let's focus on our specific function: f(x) = log₃(2x-3). Remember our golden rule? The argument of a logarithm must be positive. In our case, the argument is (2x-3). So, to find the domain, we need to ensure that this expression is strictly greater than zero. Mathematically, this translates to the following inequality:

2x - 3 > 0

This inequality is the heart of our problem. It tells us exactly what condition 'x' needs to satisfy for our function to be defined. Solving this inequality will give us the domain of f(x). This restriction arises directly from the fundamental properties of logarithmic functions, which require positive arguments to yield real values. Understanding this constraint is crucial for correctly identifying the domain and avoiding undefined function values. The argument (2x-3) represents the input to the logarithmic function, and its positivity ensures that the function operates within its valid range. This step is not merely an algebraic manipulation; it is a direct application of the logarithmic function's definition, making it a cornerstone in finding the solution. By focusing on the argument, we effectively address the core requirement for the function's existence, which is to produce real and meaningful outputs. Therefore, the inequality 2x - 3 > 0 serves as the crucial link between the function's definition and the allowable values of 'x'.

Solving the Inequality 2x - 3 > 0

Alright, let's get our hands dirty and solve this inequality! It's actually quite straightforward. Our goal is to isolate 'x' on one side of the inequality. Here’s how we do it:

  1. Add 3 to both sides: This eliminates the -3 on the left side.

    2x - 3 + 3 > 0 + 3

    Which simplifies to:

    2x > 3

  2. Divide both sides by 2: This isolates 'x'.

    2x / 2 > 3 / 2

    Which gives us:

    x > 3/2

Boom! We've solved the inequality. This tells us that 'x' must be greater than 3/2 for the function f(x) = log₃(2x-3) to be defined. Solving the inequality involves applying basic algebraic operations while maintaining the inequality's direction. Adding 3 to both sides is a direct application of the addition property of inequalities, and dividing by 2 (a positive number) preserves the inequality sign. This systematic approach ensures that each step is mathematically sound, leading to the correct solution. The result, x > 3/2, provides a clear condition for the domain: 'x' values must be strictly greater than 3/2. This condition is not just an algebraic result but a fundamental requirement dictated by the nature of logarithmic functions, emphasizing the importance of understanding the underlying mathematical principles. By isolating 'x', we've effectively defined the permissible range of input values for the function, paving the way for expressing the domain in interval notation.

Expressing the Domain in Interval Notation

So, we know that x > 3/2. That's great, but mathematicians often like to express domains using interval notation. It's a concise and clear way to show the range of values that 'x' can take. In interval notation, we use parentheses and brackets to indicate whether the endpoints of an interval are included or excluded.

  • A parenthesis '(' or ')' means the endpoint is not included.
  • A bracket '[' or ']' means the endpoint is included.

Since x > 3/2, 'x' can be any number greater than 3/2, but it cannot be equal to 3/2. Therefore, we use a parenthesis to exclude 3/2. And because 'x' can go infinitely large, we use infinity (∞) as the upper bound. Infinity always gets a parenthesis because it’s not a specific number that can be included.

So, the domain of f(x) = log₃(2x-3) in interval notation is:

(3/2, ∞)

This notation tells us, in a neat package, all the possible values of 'x' that make our function happy. Expressing the domain in interval notation offers a clear and concise way to communicate the range of valid input values. The use of parentheses around 3/2 indicates that this value is excluded from the domain, reflecting the strict inequality x > 3/2. The infinity symbol (∞) represents that the domain extends indefinitely in the positive direction, a common feature when dealing with inequalities of this nature. Interval notation is a standard convention in mathematics for representing intervals on the real number line, providing a convenient shorthand for expressing sets of numbers. It enhances clarity and precision when discussing domains and ranges of functions, making it an essential tool in mathematical communication. This notation not only specifies the limits of the domain but also provides a visual representation of the set of allowable 'x' values, facilitating a deeper understanding of the function's behavior.

Wrapping It Up

And there you have it, guys! We've successfully found the largest domain where the function f(x) = log₃(2x-3) is defined. We started by understanding the fundamental restriction of logarithmic functions (positive arguments only), set up an inequality, solved it, and then expressed our answer in interval notation. Remember, the key to these problems is always the argument of the logarithm. Make sure it's greater than zero, and you're golden!

Finding the domain of a function is a foundational skill in mathematics, critical for understanding where a function is valid and where it is not. The process we followed—identifying the restriction, setting up an inequality, solving it, and expressing the solution in interval notation—is a general method applicable to various logarithmic functions. This systematic approach not only yields the correct answer but also builds a deeper understanding of the underlying principles. By mastering the technique of determining domains, you equip yourself with a powerful tool for analyzing and working with mathematical functions across different contexts. Furthermore, understanding the domain of a function is essential for graphing and interpreting its behavior, as it defines the region on the coordinate plane where the function exists. So, the ability to find domains is not just an academic exercise but a practical skill that enhances your overall mathematical proficiency.

I hope this breakdown was helpful! Keep practicing, and you'll become a domain-finding pro in no time. Happy calculating! Remember, the more you practice these types of problems, the more comfortable and confident you'll become. Try tackling different logarithmic functions with varying arguments to solidify your understanding. Each function presents a unique challenge, and by working through them, you'll develop a robust problem-solving skillset. Don't hesitate to review the fundamental properties of logarithmic functions whenever needed, as they form the bedrock for all domain-related calculations. Moreover, consider exploring the graphical representation of logarithmic functions to visually reinforce the concept of the domain. The graph clearly shows the range of 'x' values for which the function is defined, providing an intuitive understanding of the domain. So, keep exploring, keep practicing, and embrace the world of mathematical functions!