Domain Of Logarithmic Function: F(x) = Log₉(x)

Understanding the domain of logarithmic functions is crucial for anyone studying mathematics, especially when dealing with functions and their properties. In this comprehensive guide, we'll explore what a logarithmic function is, discuss its key characteristics, and then dive deep into determining the domain of the specific function $F(x) = \log_9 x$. By the end of this article, you'll have a solid grasp of how to identify the domain of logarithmic functions, ensuring you can confidently tackle similar problems in the future.

Understanding Logarithmic Functions

To effectively determine the domain of $F(x) = \log_9 x$, it's essential to first understand the basic concept of logarithmic functions. A logarithmic function is the inverse of an exponential function. In simpler terms, if we have an exponential equation like $9^y = x$, we can rewrite it in logarithmic form as $y = \log_9 x$. The base of the logarithm (in this case, 9) is the number that is raised to a power to get the argument (x).

Key Characteristics of Logarithmic Functions:

1. Base: Logarithmic functions have a base, which is a positive real number not equal to 1. In the function $F(x) = \log_9 x$, the base is 9.
2. Argument: The argument of a logarithm is the value for which we are finding the logarithm. In $F(x) = \log_9 x$, the argument is x.
3. Inverse Relationship: Logarithmic functions are inverses of exponential functions, meaning they "undo" each other.

Why is the Domain of Logarithmic Functions Restricted?

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For logarithmic functions, the domain is restricted because you can only take the logarithm of positive numbers. This restriction arises from the very definition of a logarithm as the inverse of an exponential function. Consider the exponential function $y = a^x$, where 'a' is a positive base. No matter what value you assign to 'x', 'y' will always be positive. Consequently, when you invert this relationship to define the logarithm, you can only input positive values into the logarithm.

To illustrate this further, let's think about what a logarithm is actually asking. The expression $\log_a x$ asks the question: "To what power must we raise 'a' to get 'x'?" If 'x' is negative or zero, there is no real number that we can raise 'a' to in order to get 'x'. For example, there is no real number 'y' such that $2^y = -4$ or $2^y = 0$. Therefore, the logarithm is undefined for non-positive arguments.

In summary, the domain of logarithmic functions is restricted because logarithms are only defined for positive arguments. This restriction is a direct consequence of the inverse relationship between logarithmic and exponential functions and the fact that exponential functions always produce positive values.

Determining the Domain of F(x) = log₉(x)

Now, let's focus on the given function, $F(x) = \log_9 x$. The key to finding the domain is to remember that the argument of a logarithm must be strictly greater than zero. In other words, we can only take the logarithm of positive numbers. Therefore, for $F(x) = \log_9 x$ to be defined, x must be greater than 0.

Mathematically, this can be expressed as:

$x > 0$

This inequality tells us that the domain of the function $F(x) = \log_9 x$ consists of all real numbers greater than zero. In interval notation, this is represented as $(0, \infty)$.

Why Other Options Are Incorrect

To reinforce our understanding, let's briefly examine why the other options provided are incorrect:

• $x \geq 0$: This option includes zero in the domain, which is not allowed for logarithmic functions. The logarithm of zero is undefined.
• $x < 0$: This option includes negative numbers, which are also not allowed as arguments for logarithmic functions.
• All real numbers: This option is too broad, as it includes both negative numbers and zero, which are not valid arguments for logarithmic functions.

Practical Examples

Let's consider a few examples to solidify our understanding.

1. F(2) = log₉(2): Since 2 is greater than 0, $F(2)$ is defined. We can calculate its value using a calculator or other tools.
2. F(1) = log₉(1): Since 1 is greater than 0, $F(1)$ is defined. In fact, $\log_9(1) = 0$ because $9^0 = 1$.
3. F(-1) = log₉(-1): Since -1 is less than 0, $F(-1)$ is undefined. We cannot take the logarithm of a negative number.
4. F(0) = log₉(0): Since 0 is not greater than 0, $F(0)$ is undefined. We cannot take the logarithm of zero.

These examples clearly demonstrate that the domain of $F(x) = \log_9 x$ is restricted to positive real numbers.

Conclusion

In summary, the domain of the logarithmic function $F(x) = \log_9 x$ is all real numbers greater than 0. This is because the argument of a logarithm must always be positive. Therefore, the correct answer is:

C. $x > 0$

Understanding the domains of functions is a fundamental concept in mathematics. By recognizing the restrictions on logarithmic functions, you can accurately determine their domains and avoid common errors. Always remember that the argument of a logarithm must be strictly greater than zero. With this knowledge, you'll be well-equipped to tackle a wide range of problems involving logarithmic functions.

For further reading on logarithmic functions and their properties, you can visit Khan Academy's section on logarithms. This resource provides additional explanations, examples, and practice problems to help you master this important topic.