Domain And Range Of F(x) = Log X - 5 A Comprehensive Guide

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Hey guys! Let's dive into the fascinating world of logarithmic functions, specifically focusing on how to determine the domain and range. Today, we're going to dissect the function f(x) = log x - 5. Understanding the domain and range is crucial for grasping the behavior and characteristics of any function, and logarithmic functions have some unique properties that make them particularly interesting. So, buckle up, and let's get started!

What are Domain and Range?

Before we jump into our specific function, let's quickly define what we mean by domain and range. Think of a function as a machine: you put something in (the input), and the machine spits something out (the output).

  • Domain: The domain is the set of all possible inputs (x-values) that you can feed into the function without causing any mathematical mayhem. In other words, it's all the values of x for which the function is defined.
  • Range: The range, on the other hand, is the set of all possible outputs (y-values or f(x) values) that the function can produce. It's all the values you get out of the function when you plug in all the valid inputs from the domain.

Delving into Logarithmic Functions

Now that we're clear on domain and range, let's talk about logarithmic functions in general. The logarithmic function is the inverse of the exponential function. If you have an exponential function like y = a^x, where a is the base, then the corresponding logarithmic function is written as x = log_a(y). In simpler terms, the logarithm answers the question: "To what power must we raise the base 'a' to get y?"

The most common logarithmic function you'll encounter is the common logarithm, which has a base of 10. When the base isn't explicitly written, it's generally assumed to be 10. So, log x is the same as log_10(x). Our function, f(x) = log x - 5, uses the common logarithm.

Logarithmic functions have a vertical asymptote at x = 0. This is a critical point to remember! A vertical asymptote is a vertical line that the graph of the function approaches but never actually touches. In the case of logarithmic functions with a base greater than 1 (like our common logarithm with base 10), the function is only defined for positive values of x. This is because you can't raise a positive number (the base) to any power and get a non-positive result (zero or a negative number). This restriction directly impacts the domain of logarithmic functions.

The graph of a logarithmic function increases slowly as x increases, but it spans all possible y-values. This means that while the x-values are restricted, the y-values are not, which is a key to understanding the range.

Analyzing f(x) = log x - 5: Finding the Domain

Alright, let's get our hands dirty with our function, f(x) = log x - 5. The core of this function is the log x part. As we discussed, logarithmic functions (with a base greater than 1) are only defined for positive arguments. The argument is the expression inside the logarithm, which in this case is simply 'x'.

Therefore, to find the domain, we need to ensure that the argument, x, is strictly greater than zero. We can write this as an inequality:

x > 0

This inequality tells us that the domain of f(x) = log x - 5 is all real numbers greater than zero. We can represent this in interval notation as (0, ∞). This means the function accepts any positive number as input, but it's not defined for zero or negative numbers. Think about it: you can't ask, "To what power must we raise 10 to get a negative number or zero?" There's no answer!

So, the domain of f(x) = log x - 5 is x > 0. This is a crucial piece of the puzzle.

Analyzing f(x) = log x - 5: Unveiling the Range

Now that we've conquered the domain, let's tackle the range. The range is the set of all possible output values (y-values) that our function can produce. To determine the range, we need to consider the behavior of the logarithmic function and the effect of the "- 5" part of our function.

The basic logarithmic function, log x, can take on any real number as its output. As x gets closer and closer to zero (from the positive side), log x approaches negative infinity. As x gets larger and larger, log x also increases, albeit slowly, and approaches positive infinity. This means that the range of log x itself is all real numbers.

Now, let's consider the "- 5" in our function, f(x) = log x - 5. This "- 5" represents a vertical shift. It shifts the entire graph of the log x function downwards by 5 units. However, this vertical shift doesn't change the range! Since the original log x function can produce any real number, shifting it up or down will still result in the function being able to produce any real number.

Imagine the graph stretching infinitely upwards and downwards. Shifting it down by 5 units simply moves the entire infinite stretch; it doesn't limit the possible y-values. Therefore, the range of f(x) = log x - 5 is also all real numbers.

We can represent the range using interval notation as (-∞, ∞), which signifies all real numbers.

Putting It All Together: The Domain and Range of f(x) = log x - 5

We've successfully dissected the function f(x) = log x - 5 and determined both its domain and its range. Let's recap:

  • Domain: x > 0 (all real numbers greater than zero)
  • Range: All real numbers

This corresponds to option A in your initial choices. So the answer is A. domain: x > 0; range: all real numbers.

Visualizing the Function

It's often helpful to visualize functions to solidify our understanding. If you were to graph f(x) = log x - 5, you would see a curve that starts very close to the y-axis (but never touches it) and gradually increases as x increases. The graph extends infinitely upwards and downwards, confirming our range of all real numbers. The fact that the graph exists only to the right of the y-axis confirms our domain of x > 0.

Key Takeaways

  • Logarithmic functions have a restricted domain: The argument of the logarithm must be positive.
  • The range of a basic logarithmic function is all real numbers.
  • Vertical shifts do not affect the range.
  • Understanding the base of the logarithm is crucial for determining the function's behavior.

Practice Makes Perfect

To truly master the concepts of domain and range, practice is key! Try finding the domain and range of other logarithmic functions, such as f(x) = log(x + 2), f(x) = 2log(x), or f(x) = log(-x). Pay close attention to how transformations (shifts, stretches, and reflections) affect the domain and range.

By working through these examples, you'll develop a strong intuition for how logarithmic functions behave and how to confidently determine their domain and range. Keep exploring, keep practicing, and you'll become a domain and range pro in no time! Remember, math is an adventure – enjoy the journey!

I hope this comprehensive explanation has helped you understand the domain and range of f(x) = log x - 5. If you have any further questions, don't hesitate to ask! Keep learning and keep exploring the wonderful world of mathematics!