Finding The Domain Of F(x) = Log5(x+9) A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of logarithmic functions and tackling a common question: how to determine the domain of a logarithmic function. Specifically, we'll be dissecting the function ${f(x) = \log_5(x+9)}$. It might seem a bit daunting at first, but trust me, with a clear understanding of the underlying principles, it becomes quite straightforward. So, let's put on our math hats and get started!

Understanding Logarithmic Functions and Their Domains

Before we jump directly into solving for the domain of our specific function, ${f(x) = \log_5(x+9)}$, it's crucial to establish a strong foundation by understanding what logarithmic functions are and the key restrictions they carry regarding their domains. Logarithmic functions, at their core, are the inverse operations of exponential functions. Think of it this way: if exponentiation is like repeatedly multiplying a number by itself, logarithms are like asking, "How many times do I need to multiply this number by itself to get a specific result?" For instance, if we have the exponential equation ${5^2 = 25}$, the equivalent logarithmic form would be ${\log_5(25) = 2}$, which reads as "the logarithm base 5 of 25 is 2." This simply means that 5 raised to the power of 2 equals 25.

Now, where does the domain come into play? The domain of a function, as you might already know, is the set of all possible input values (often represented by ${x}$) for which the function produces a valid output. Logarithmic functions have a critical restriction on their domain: they are only defined for positive arguments. This restriction stems directly from the nature of exponential functions. Consider the general form of a logarithmic function: ${f(x) = \log_b(x)}$, where ${b}$ is the base (a positive number not equal to 1). The logarithm asks, "To what power must we raise ${b}$ to obtain ${x}$?" If ${x}$ is zero or negative, there is no real number power to which we can raise ${b}$ to get that result. Think about it – a positive number raised to any power will always be positive. We can never get zero or a negative number. This fundamental restriction is the key to finding the domain of any logarithmic function.

In essence, when dealing with logarithmic functions, you must always ensure that the argument (the expression inside the logarithm) is strictly greater than zero. This is the golden rule, the guiding principle that will lead you to the correct domain. This might seem a bit abstract right now, but as we apply this concept to our example function, ${f(x) = \log_5(x+9)}$, the practical implications will become crystal clear. Remember this: positive arguments are the lifeblood of logarithmic functions. Without them, the function simply cannot exist within the realm of real numbers.

Solving for the Domain of ${f(x) = \log_5(x+9)}$

Alright, now that we've solidified our understanding of logarithmic functions and their inherent domain restrictions, let's get our hands dirty and determine the domain of our specific function: ${f(x) = \log_5(x+9)}$. Remember our golden rule? The argument of the logarithm, the expression inside the parentheses, must be strictly greater than zero. In this case, our argument is ${x + 9}$. So, to find the domain, we need to solve the following inequality:

${x + 9 > 0}$

This inequality is quite simple to solve. Our goal is to isolate ${x}$ on one side of the inequality. To do this, we can subtract 9 from both sides of the inequality. Remember, when dealing with inequalities, performing the same operation on both sides maintains the inequality (unless you're multiplying or dividing by a negative number, which we aren't doing here). So, subtracting 9 from both sides, we get:

${x + 9 - 9 > 0 - 9}$

This simplifies to:

${x > -9}$

And there you have it! We've solved the inequality and found the condition that ${x}$ must satisfy for the function ${f(x) = \log_5(x+9)}$ to be defined. The solution ${x > -9}$ tells us that the domain of the function consists of all real numbers greater than -9. It's important to note the "greater than" part – ${x}$ cannot be equal to -9 because that would make the argument of the logarithm zero, which is not allowed. Remember, only positive arguments are permitted for logarithms.

Now, let's express this solution in interval notation, which is a common and concise way to represent sets of numbers. Interval notation uses parentheses and brackets to indicate the endpoints of an interval. A parenthesis, ${(}$ or ${)}$, indicates that the endpoint is not included in the interval, while a bracket, ${) or ${}$}$, indicates that the endpoint is included. Since our solution is ${x > -9}$, we want to represent all numbers greater than -9, but not including -9 itself. Therefore, the interval notation for the domain of ${f(x) = \log_5(x+9)}$ is:

${(-9, \infty)}$

This notation reads as "the open interval from -9 to infinity." The parenthesis next to -9 signifies that -9 is not included, and the infinity symbol indicates that the interval extends indefinitely to the right. This is the most accurate and standard way to represent the domain of our logarithmic function.

Choosing the Correct Answer and Wrapping Up

Okay, we've successfully navigated the world of logarithmic functions, tackled the inequality, and expressed the domain in interval notation. Now, let's circle back to the original question and select the correct answer from the options provided. The options were:

• A. ((0, \infty)]
• B. ((-9, \infty)]
• C. ((9, \infty)]
• D. ((-\infty, \infty)]

Based on our step-by-step solution, we determined that the domain of ${f(x) = \log_5(x+9)}$ is all real numbers greater than -9, which is represented in interval notation as ${(-9, \infty)}$. Therefore, the correct answer is:

• B. ((-9, \infty)]

Fantastic! We've successfully identified the domain of the given logarithmic function. Pat yourselves on the back, guys! You've not only learned how to solve this specific problem but also gained a deeper understanding of the fundamental principles governing the domains of logarithmic functions. Remember, the key takeaway is that the argument of a logarithm must always be positive. This principle, combined with the ability to solve inequalities, will empower you to tackle a wide range of domain-related problems involving logarithmic functions.

To recap, we started by establishing a solid foundation in logarithmic functions and their domain restrictions, emphasizing the crucial requirement of a positive argument. We then applied this knowledge to our example function, ${f(x) = \log_5(x+9)}$, by setting up and solving the inequality ${x + 9 > 0}$. This led us to the solution ${x > -9}$, which we then expressed in the concise and standard interval notation as ${(-9, \infty)}$. Finally, we confidently selected the correct answer from the provided options.

This journey through the domain of logarithmic functions highlights the importance of understanding the underlying principles and applying them systematically. Math, at its core, is about building a strong foundation of knowledge and using that foundation to solve progressively more complex problems. Keep practicing, keep exploring, and keep challenging yourselves. The world of mathematics is vast and rewarding, and with a little bit of effort and the right approach, you can conquer any challenge that comes your way. So, until next time, happy problem-solving! And remember, always keep those arguments positive!