Mathematical Proof: Real Numbers with Exponential but No Logarithmic Functions

Exploring the nature of mathematical functions can sometimes be a fascinating journey, especially when delving into the intricate relationships between different types of functions. One intriguing question that arises is whether there exists a real number (x) such that the exponential function (e^x) is well-defined but the logarithmic function (log(x)) is not. This article aims to explore this question in depth, providing a clear mathematical proof and understanding the implications.

Introduction to Real Numbers and Functions

Before diving into the main topic, let's briefly review some fundamental concepts. Real numbers are a complete ordered field, meaning they include all rational and irrational numbers. Functions, particularly the exponential and logarithmic functions, play a crucial role in mathematics and have numerous applications in science and engineering. The exponential function is of the form (e^x), where (e) is the base of the natural logarithm, approximately equal to 2.71828. The logarithmic function, on the other hand, is defined as the inverse of the exponential function. It is denoted as (log(x)) or (ln(x)), and both functions have distinct domains and ranges.

Defining Exponential and Logarithmic Functions

The exponential function (e^x) is defined for all real numbers (x). It is a strictly increasing function and passes through the point ((0,1)). Mathematically, it is defined as:

(e^x sum_{n0}^{infty} frac{x^n}{n!})

This series converges for all real (x), ensuring that (e^x) is well-defined everywhere on the real number line.

The logarithmic function (log(x)) or (ln(x)) is the inverse of the exponential function (e^x). However, the logarithm is only defined for positive real numbers (x). This is because there is no real number (x) such that (e^x y) for (y leq 0). Therefore, the domain of (log(x)) is strictly positive real numbers, (x > 0).

The Existence of Real Numbers with Exponential but No Logarithmic Functions

Given the definitions of the exponential and logarithmic functions, we need to address the possibility of a real number (x) such that (e^x) is defined but (log(x)) is not. From the domain of the logarithmic function, it is immediately clear that (log(x)) is not defined for any (x leq 0). Hence, any real number (x) where (x leq 0) would satisfy the condition that (e^x) is defined but (log(x)) is not. A simple and intuitive example is (x 0).

Example: Let (x 0).

(e^0 1), which is a well-defined real number. The exponential function (e^x) is indeed defined for (x 0). (log(0)) is undefined because there is no real number (y) such that (e^y 0). This follows directly from the definition of the exponential function, as it never reaches zero.

Therefore, (x 0) is a concrete example of a real number where the exponential function (e^x) is defined but the logarithmic function (log(x)) is not. This confirms the existence of such real numbers.

Conclusion

In conclusion, the existence of real numbers (x) such that (e^x) is defined but (log(x)) is not is mathematically proven. The number (x 0) serves as an example, illustrating the clear distinction between the domains of the exponential and logarithmic functions. Understanding these nuances is crucial for a deeper comprehension of mathematical functions and their applications.

Key Takeaways

Exponential functions are defined for all real numbers (x). Logarithmic functions are only defined for positive real numbers (x > 0). A real number (x leq 0) satisfies the condition where (e^x) is defined but (log(x)) is not.

Frequently Asked Questions

What is the difference between exponential and logarithmic functions?

The exponential function (e^x) grows rapidly and is defined for all real numbers (x). The logarithmic function (log(x)) is the inverse of the exponential function and is defined only for positive real numbers (x > 0).

Can a real number be both positive and undefined in a logarithmic function?

By definition, a logarithmic function (log(x)) cannot have a real number (x leq 0) as its argument since the exponential function (e^x) never equals a non-positive value. Thus, any positive real number (x > 0) is a valid argument for a logarithmic function, while any non-positive number (x leq 0) is not.