Problem 22 - Discrete Mathematics with Applications

Question

Prove by contradiction, where \(p\) is a prime number. \(\log _{10} 2\) is an irrational number.

Step-by-Step Solution

Answer

Assuming that \(\log_{10} 2\) is a rational number and can be expressed as a fraction \(\frac{a}{b}\), we convert it to an exponential form: \(10^{\frac{a}{b}}=2\). By finding a common base and utilizing logarithm properties, we get the equation \(a \log _2 10 = 1\), which leads to \(\log _2 10 = \frac{1}{a}\). This equation contradicts the known fact that \(\log _2 10\) is an irrational number, proving by contradiction that \(\log_{10} 2\) is indeed an irrational number.

Step 1: State the Assumption

Let's start by assuming that \(\log _{10} 2\) is a rational number. This means that \(\log _{10} 2\) can be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\).

Step 2: Convert to Exponential Form

Using the definition of logarithms, we can rewrite \(\log _{10} 2\) as an exponential equation. Since \(\log _{10} 2 = \frac{a}{b}\), we can write this as: \[10^{\frac{a}{b}} = 2\]

Step 3: Find a Common Base

In order to simplify the equation, we need to express both sides of the equation with a common base. Since \(2\) and \(10\) are both powers of \(2\), we will use base \(2\). This yields: \[2^{b\log _2 10^{\frac{a}{b}}} = 2\]

Step 4: Utilize Logarithm Properties

Apply the power rule of logarithms to the left-hand side of the equation: \[2^{b(\frac{a}{b} \log _2 10)} = 2\] The \(b\) in the exponent cancels out, leaving: \[2^{a \log _2 10} = 2\]

Step 5: Set up Equation for Contradiction

For contradiction, we want to find an equation that equates two integers. From the above simplified equation, we see that: \[a \log _2 10 = 1\] Now, let's isolate \(\log _2 10\): \[\log _2 10 = \frac{1}{a}\] Since \(a\) is an integer according to our initial assumption, the right-hand side of this equation is a rational number.

Step 6: Contradiction

In step 5, we determined that \(\log _2 10\) is a rational number. However, it is well-known that \(\log _2 10\) is an irrational number, as we cannot express it as a simple fraction. This is a contradiction to our assumption that \(\log _{10} 2\) is a rational number, leading us to the conclusion that the assumption is false. This proves by contradiction that \(\log_{10} 2\) is an irrational number.

Key Concepts

Irrational Numbers

An irrational number is a type of real number that cannot be expressed as a simple fraction of two integers. This means that its decimal expansion is non-terminating and non-repeating; it goes on forever without a recurring pattern. Common examples of irrational numbers include \( \pi \) and \( \sqrt{2} \). The fact that the decimal expansion doesn’t repeat itself means you can't precisely represent an irrational number as a fraction, which is essential in distinguishing them from rational numbers.

Understanding irrational numbers is critical when learning about logarithms. The exercise asks to prove that \( \log_{10} 2 \) is irrational. The notion that a logarithm can be irrational is sometimes counterintuitive, since logarithmic functions often arise from real-world measurements which are typically rational. However, real-world measures can lead to irrational numbers, just as in the case of logarithms.

Logarithms

A logarithm is the inverse operation to exponentiation, answering the question: 'To what exponent must the base be raised, to produce a certain number?' For example, \( \log_{10} 100 = 2 \) because \( 10^2 = 100 \). Logarithms have a variety of properties that make them useful in simplifying complex calculations, particularly those involving exponential relationships.

Key properties include the product rule \( \log_{b}(MN) = \log_{b}(M) + \log_{b}(N) \), the quotient rule \( \log_{b}(\frac{M}{N}) = \log_{b}(M) - \log_{b}(N) \), and the power rule \( \log_{b}(M^n) = n \log_{b}(M) \). In the given exercise, the use of logarithms allows the transformation of an exponential equation into a form that can be analyzed for its rationality or irrationality. In proving that \( \log_{10} 2 \) is irrational by contradiction, these properties of logarithms are essential.

Exponential Equations

Exponential equations involve variables in the exponents and are of the form \( b^x = y \), where \( b \) is the base and \( y \) is the result of the exponentiation. They are fundamental in modeling growth and decay in diverse fields like biology, finance, and physics. Solving exponential equations often requires manipulation to express both sides of the equation with a common base, which enables us to simplify and solve the equation by equating exponents.

The step-by-step solution provided demonstrates this process by showing how one can express \( 10^{\frac{a}{b}} = 2 \) as an equation with a common base, facilitating the use of logarithms to find a contradiction. The logic behind the transformation from exponential form to logarithmic form is crucial in proving that certain numbers, such as \( \log_{10} 2 \) in this case, are indeed irrational numbers.