Proving the Irrationality and Transcendence of Logarithmic Expressions

When examining the logarithmic expression log_2{3}, it is not immediately clear whether it is a rational or irrational number. This article will explore the proof that log_2{3} is actually an irrational number and, more impressively, a transcendental number. Using fundamental concepts in number theory and some powerful theorems, we will walk through the steps to reach these conclusions.

Proving Rationality Leads to Absurdity

Assume for contradiction that log_2{3} is a rational number. This implies that there exist integers m and n such that:

log_2{3} frac{m}{n}

By the definition of the logarithm, we can write 3 2^{frac{m}{n}}. Raising both sides to the power of n, we get:

3^n 2^m

According to the Fundamental Theorem of Arithmetic, any integer has a unique prime factorization. Therefore, for the equation 3^n 2^m to hold, m must be a multiple of 3 and n must be a multiple of 2. However, this is impossible as the left-hand side (LHS) is odd and the right-hand side (RHS) is even. Thus, our assumption that log_2{3} is rational leads to a contradiction. Hence:

**log_2{3} is irrational.**

Applying the Gelfond-Schneider Theorem

Next, we will investigate whether 2^{log_2{3}} is algebraic or transcendental. The expression 2^{log_2{3}} 3 is clearly algebraic because it is an integer and thus a solution to a polynomial equation (e.g., x - 3 0). According to the contrapositive of the Gelfond-Schneider theorem, if 2 and log_2{3} are algebraic and log_2{3} is irrational, then 2^{log_2{3}} must be transcendental. However, since 2^{log_2{3}} 3 is clearly algebraic, we conclude that log_2{3} must be transcendental:

**log_2{3} is transcendental.**

Generalizing the Result

The argument we have used for log_2{3} can be generalized to any positive integers a and b. We have shown that log_b a is either rational or transcendental. Therefore, for the logarithm of any positive integer base with respect to another positive integer, the result is either a rational number or a transcendental number, but not an irrational algebraic number:

For any positive integers a and b, the expression log_b a is either rational or transcendental.

Conclusion

In conclusion, we have rigorously proven that log_2{3} is irrational and, more specifically, transcendental. The methods used in this proof involve the manipulation of logarithmic expressions, the Fundamental Theorem of Arithmetic, and the powerful Gelfond-Schneider Theorem. This example demonstrates the importance of these concepts in number theory and their practical applications in proving irrationality and transcendence.