91Ó°ÊÓ

Logarithms are a fundamental concept in mathematics, often used to solve equations involving exponential growth or decay. The logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number.
For example, if we have a logarithm with base 2, noted as \(\log_{2}\), we are asking the question: "To what power must 2 be raised, to achieve a specific number?" Hence, \(\log_{2} 8 = 3\) because \(2^3 = 8\).
This basic idea allows us to switch between multiplication (exponentiation) and addition (logarithm), making complex calculations simpler.

Key properties of logarithms include:

• Product Rule: \(\log_b(MN) = \log_b(M) + \log_b(N)\)
• Quotient Rule: \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\)
• Power Rule: \(\log_b(M^k) = k \cdot \log_b(M)\)

These properties are incredibly useful when rewriting or simplifying logarithmic expressions, as shown in our exercise where \(\log_{2} 3\) was assumed to be rational for a proof by contradiction.

Proof by contradiction is a classical method used in mathematics to establish the truth of a statement. The idea is to assume the opposite of what you want to prove, then show that this assumption leads to a contradiction, implying the original statement must be true.
Here's a simple breakdown of this process:

• Assumption: Start by assuming the opposite of the statement you want to prove.
• Logical Deduction: Use logical reasoning and known facts to explore the consequences of this assumption.
• Identify Contradiction: Find a contradiction—a situation that conflicts with known facts, rules, or assumptions.

A classic example is proving that \(\sqrt{2}\) is irrational. By assuming that \(\sqrt{2}\) is rational, you eventually reach the paradox of a reduced fraction containing two even integers, contradicting the definition of rational numbers.
In our exercise, assuming \(\log_{2} 3\) is rational leads to the impossible equation \(2^p = 3^q\), a contradiction because no power of 2 can equal a power of 3.

Prime numbers are integers greater than 1 that have no divisors other than 1 and themselves. They are the building blocks of number theory because any positive integer is a product of primes, known as its prime factorization.
The uniqueness of prime factorization ensures that different numbers cannot have the same set of prime factors.
For example:

• 2, 3, 5, and 7 are all prime numbers.
• A number like 60 has a prime factorization of 2² × 3 × 5.

Prime numbers are crucial in our exercise, where we used them to demonstrate that \(2^p = 3^q\) (derived from \(\log_{2} 3\) being rational) cannot be true. Each side of the equation involves different bases (2 and 3), and since both are primes, their powers will never equate. This impossibility highlights the foundational role of primes in proving irrationality in certain contexts.