Problem 119 - Calculus and its applications

Question

To prove Properties \(P 1, P 2, P 3,\) and \(P 7\) of Theorem \(3,\) let \(X=\log _{a} M\) and \(Y=\log _{a} N,\) and give reasons for the steps listed in Exercises. $$ \begin{array}{l} M=a^{X} \text { and } N=a^{Y} \\ \text { so } M N=a^{X} \cdot a^{Y}=a^{X+Y} \\ \text { Thus, } \log _{a}(M N)=X+Y \\ =\log _{a} M+\log _{a} N . \end{array} $$

Step-by-Step Solution

Answer

The property is proven: \(\log_a(MN) = \log_a M + \log_a N\).

Step 1: Express Variables

Define the variables given in the problem. Let \( X = \log_{a} M \) and \( Y = \log_{a} N \). This means, by definition of logarithms, \( M = a^X \) and \( N = a^Y \). This establishes a relationship between the logs and their corresponding exponential forms.

Step 2: Multiply the Exponential Forms

Calculate the product of \( M \) and \( N \) using their exponential forms. Since \( M = a^X \) and \( N = a^Y \), the product is \( MN = a^X times a^Y \). Use the rule of exponents which states that \( a^m times a^n = a^{m+n} \). Therefore, \( MN = a^{X+Y} \).

Step 3: Apply Logarithm to Product

Take the logarithm to the base \( a \) of both sides of the equation \( MN = a^{X+Y} \). By the property of logarithms, the left hand side becomes \( \log_a(MN) \) and the right hand side turns into \( X + Y \). Therefore, \( \log_a(MN) = \log_a(a^{X+Y}) \).

Step 4: Simplify Using Logarithm Properties

As \( \log_a(a^{X+Y}) = X+Y \), use the definitions \( X = \log_a M \) and \( Y = \log_a N \) to get \( X + Y = \log_a M + \log_a N \). Hence, \( \log_a(MN) = \log_a M + \log_a N \). This proves the property of logarithms that \( \log_a(MN) = \log_a M + \log_a N \).

Key Concepts

Exponential Functions

Exponential functions are fundamental in mathematics, particularly when dealing with growth and decay models. An exponential function has the form \(f(x) = a^x\) where \(a\) is a constant and \(x\) is the exponent. These functions are characterized by their rapid increase or decrease.

• The base \(a\) in an exponential function must be a positive real number, and it is typically greater than 1 for growth models or between 0 and 1 for decay models.
• The exponent \(x\) can be any real number and determines the function's value for any given \(a\).

The relationship between exponential functions and logarithms is crucial. For instance, given \(y = a^x\), taking the logarithm base \(a\) of both sides gives \(x = \log_a(y)\), showing how logarithms can be used to solve for variables in exponential equations.
Exponent rules, such as \(a^m \times a^n = a^{m+n}\), are essential in manipulating and simplifying expressions involving exponential functions, as seen in the textbook problem.

Logarithmic Identities

Logarithms are the inverses of exponential functions and are used to solve equations where the unknown variable is in an exponent. Some essential properties of logarithms, called logarithmic identities, make calculations more manageable:

• Product Property: \(\log_a(MN) = \log_a M + \log_a N\)
• Quotient Property: \(\log_a\left(\frac{M}{N}\right) = \log_a M - \log_a N\)
• Power Property: \(\log_a(M^b) = b \cdot \log_a M\)

These identities are derived from the properties of exponents. The product property, for example, stems from the rule \(a^m \times a^n = a^{m+n}\), allowing multiplication of numbers in a logarithmic form to convert to addition. Understanding these identities makes it easier to work with complex logarithmic equations.
The step-by-step solution in the problem uses the product property of logarithms to prove that \(\log_a(MN) = \log_a M + \log_a N\). Recognizing and applying these identities effectively can simplify and solve a wide range of mathematical problems involving logarithms.

Mathematical Proof

A mathematical proof is a logical argument that verifies the truth of a mathematical statement. It is an essential part of mathematics, ensuring that concepts and theories hold under all circumstances.
There are various methods for constructing a proof:

• Direct Proof: Use definitions and established results to demonstrate the truth of a statement directly.
• Indirect Proof: Assume the opposite of what you want to prove, and show this leads to a contradiction.
• Proof by Induction: Prove a base case, and then show that if the statement holds for an arbitrary case, it also holds for the next case.

In the exercise, a direct proof method is used to demonstrate a logarithmic identity. By starting with known definitions of logarithms and properties of exponents, each step logically follows the previous to arrive at the conclusion \(\log_a(MN) = \log_a M + \log_a N\). This careful reasoning and logical structure ensure the identity is valid.