Question

Modify the proof outlined in Exercise 84 and use property E2 for exponents to prove that \(\log _{b}(x / y)=\log _{b} x-\log _{b} y\)

Short answer

Question: Prove that \(\log_b(x / y) = \log_b x - \log_b y\). Answer: To prove this relationship, we assumed \(y_1 = \log_b x\) and \(y_2 = \log_b y\) and used the definition of logarithms with property E2 for exponents (\(b^x \cdot b^y = b^{x+y}\)). After some algebraic manipulation, we arrived at the equation \(\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y\), which completes the proof.

Step-by-step solution

Definition of Logarithm

Recall that if \(b^\alpha = x\), then \(\log_b x = \alpha\). From this definition, let's assume \(y_1 = \log_b x\) and \(y_2 = \log_b y\). Then, we have \(b^{y_1} = x\) and \(b^{y_2} = y\).

Using Property E2

By applying property E2, we get \(b^{y_1} \cdot b^{-y_2} = b^{y_1-y_2}\). Now, we will substitute the values of \(b^{y_1}\) and \(b^{y_2}\) obtained from the definition of logarithm: $$x \cdot \frac{1}{y} = b^{y_1-y_2}$$ $$\frac{x}{y} = b^{y_1-y_2}$$

Applying Logarithm

Now, apply the logarithm with base \(b\) to both sides of the equation: $$\log_b \left(\frac{x}{y}\right) = \log_b (b^{y_1-y_2})$$ By definition of logarithm, we have: $$\log_b \left(\frac{x}{y}\right) = y_1 - y_2$$

Back to Original Variables

We need to substitute back \(y_1\) and \(y_2\) with their expressions in terms of \(x\) and \(y\): $$\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$$ So, the proof is complete.

Key concepts

Definition of Logarithm

The concept of a logarithm is fundamental in understanding various mathematical and real-world phenomena. By definition, if we have an equation of the form \( b^{\text{power}} = \text{number} \), where \( b \) is the base and \( \text{number} \) is the result, the logarithm answers the question: to what power must the base \( b \) be raised to produce \( \text{number} \)? Formally, if \( b^{\text{power}} = \text{number} \), then the logarithm of \( \text{number} \) to base \( b \) is \( \text{power} \), written as \( \text{power} = \text{log}_{b}(\text{number}) \).

Understanding the definition of logarithm is pivotal when dealing with exponential equations and growth patterns, as it provides a way of 'undoing' the exponentiation process. In fields like acoustics, earthquakes' intensity scale, and computer science, logarithms help to measure quantities that have enormous ranges. In the context of the exercise, recognizing that log base \( b \) of \( x \) is the power to which \( b \) must be raised to get \( x \) is the foundation that connects to the property that allows us to prove the relationship between the logarithm of a quotient and the difference of logarithms.

Properties of Exponents

Exponents express repeated multiplication and have their own rules that simplify complex mathematical operations. These properties or rules are crucial when we are working with exponential expressions. One of the foundational properties is Property E2, the quotient of powers property, stating that for any non-zero number \( b \) and any integers \( m \) and \( n \), \( b^{m} / b^{n} = b^{m-n} \).

In layman's terms, when you divide two numbers with the same base, you simply subtract their exponents. This property underlies many operations in algebra, helping to manipulate and simplify expressions. It is also heavily used when dealing with growth and decay in exponential functions. In quantitative fields like finance, understanding these properties is essential for calculating compound interest. In the presented exercise, applying this property allows us to combine two exponentiated terms into a single term, a precursor to applying logarithms for further simplification.

Logarithmic Identities

Logarithmic identities are mathematical tools that help us simplify and evaluate expressions involving logarithms. Among these are identities that address the logarithms of products, quotients, and powers. They empower us to break down complex logarithmic expressions into simpler parts that are often easier to calculate or understand. Specifically, the logarithm of a product is the sum of the logarithms of the factors; the logarithm of a quotient is the difference of the logarithms; and the logarithm of a power is the exponent times the logarithm of the base.

These identities reflect the interconnected nature of logarithms and exponents, being essentially inverse operations. Understanding and applying these identities enable us to solve the exercise by transforming the logarithm of a quotient into a subtraction of logarithms. Through the use of these identities, students can simplify solutions to problems involving growth rates, sound intensity, and acidity in chemistry. Mastering logarithmic identities is not only about finding solutions to equations but also about developing a deeper appreciation for the elegance and consistency of mathematical relationships.