Problem 68 - Precalculus : Building Concepts and Connections

Question

In Exercises \(65-68,\) use a graphing utility with a decimal window. Graph \(f(x)=\log x-\log (x-1)\) and \(g(x)=\log \frac{x}{x-1}\) on the same set of axes. (a) What are the domains of the two functions? (b) For what values of \(x\) do these two functions agree? (c) To what extent does this pair of functions exhibit the quotient property of logarithms?

Step-by-Step Solution

Answer

The domains of the two functions are both \(x > 1\). The functions agree for all \(x > 1\). This pair of logarithmic functions fully exhibits the quotient property.

Step 1: Graph the Two Functions

Using a graphing utility, graph both functions \(f(x)=\log x-\log (x-1)\) and \(g(x)=\log \frac{x}{x-1}\).

Step 2: Determine the Domains

The domain of a function is set of all possible x-values. For a logarithmic function, the argument must be greater than zero. Therefore, for the first function, the domain is \(x > 1\), and for second function, it is also \(x > 1\).

Step 3: Find the Intersection

To find the values of \(x\) for which these two functions agree, it is necessary to set them equal to one another and solve for \(x\), or visually find the points of intersection on the graph. In this case of the logarithm functions, they are equal for all \(x > 1\).

Step 4: Analyze the Quotient Property

Comparing the graphed functions to the quotient property of logarithms, it's clear that the two functions exhibit this property exactly for all \(x > 1\). This is because the form \(f(x) = \log(M) - \log(N)\) simplifies to \(log(M/N)\) according to the quotient property, which is indeed the form of the function \(g(x)\).

Key Concepts

Graphing Utility

A graphing utility, like a graphing calculator or software, is a powerful tool for visualizing mathematical functions. When handling logarithmic functions like our pair, a graphing utility helps illustrate the behavior and characteristics of the functions across different domains. By entering the functions into the utility, you can see their graph over a selected range of x-values and understand their relationship better.
In our exercise, the two functions, \(f(x) = \log x - \log (x-1)\) and \(g(x) = \log \frac{x}{x-1}\), are graphed using a graphing utility. This allows us to see how they behave and determine their intersection points. A decimal window in the utility helps view these with precision, especially for values of \(x\) closely greater than 1.
Using a graphing utility simplifies analyzing complex functions, making it less daunting to predict intersection points and compare different functions' properties.

Domains of Functions

The domain of a function is crucial because it determines the input values the function can accept. For logarithmic functions, the argument (the value inside the log function) must be greater than zero. This rule applies to \(f(x) = \log x - \log(x-1)\) and \(g(x) = \log \frac{x}{x-1}\).
For \(f(x)\), both \(x\) and \(x-1\) need to be positive, leading us to the domain \(x > 1\). Similarly, since \(g(x)\) also involves the fraction \(\frac{x}{x-1}\), the condition \(x > 1\) holds here too. Therefore, both functions share the same domain, starting just beyond 1, where both arguments become valid.
Understanding the concept of domains helps prevent mathematical errors that occur by plugging in invalid inputs, often leading to undefined results or complex numbers.

Quotient Property of Logarithms

The quotient property of logarithms is a useful tool that simplifies expressions involving division inside logarithms. According to this property, \(\log \frac{M}{N} = \log M - \log N\). This makes it easier to handle complex logarithmic expressions.
In this exercise, the function \(f(x) = \log x - \log (x-1)\) directly showcases the quotient property when rewritten as \(\log \frac{x}{x-1}\), which is exactly the form of \(g(x)\). This property simplifies calculations and shows equivalence between these differing forms of the same mathematical idea.
Grasping the quotient property allows for transformations between different expressions, making solving logarithmic equations more efficient and simplifying expressions during algebraic manipulations.