Problem 36 - Calculus of a Single Variable

Question

Verifying Properties of Logarithms In Exercises 35 and \(36,(a)\) verify that \(f=g\) by using a graphing utility to graph \(f\) and \(g\) in the same viewing window and (b) verify that \(f=g\) algebraically. $$ f(x)=\ln \sqrt{x\left(x^{2}+1\right)}, \quad g(x)=\frac{1}{2}\left[\ln x+\ln \left(x^{2}+1\right)\right] $$

Step-by-Step Solution

Answer

The functions \(f(x)=\ln \sqrt{x\left(x^{2}+1\right)}\) and \(g(x)=\frac{1}{2}\left[\ln x+\ln\left(x^{2}+1\right)\right]\) are identical, as verified both graphically and algebraically.

Step 1: Graph the functions

Using a graphing utility, plot the functions \(f(x)=\ln \sqrt{x\left(x^{2}+1\right)}\) and \(g(x)=\frac{1}{2}\left[\ln x+\ln\left(x^{2}+1\right)\right]\). Observe that the two functions coincide, indicating that they are indeed identical, at least graphically.

Step 2: Algebraic verification - express \(f(x)\) in a comparable form to \(g(x)\)

To verify the equality algebraically, express the function \(f(x)\) in a form that can be directly compared to \(g(x)\). \n\nThe function \(f\) is currently in the form \(f(x) = \ln \sqrt{x\left(x^{2}+1\right)}\), which can be written as \(f(x) = \ln (x(x^2+1))^{1/2}\). \n\nUsing the property of logarithms \(\ln a^{n} = n\ln a\), \(f(x)\) can be written as \(f(x) = \frac{1}{2}\ln(x(x^2+1))\). \n\nAgain, applying the property \(\ln ab = \ln a + \ln b\), \(f(x)\) can be further broken down to \(f(x) = \frac{1}{2}[\ln x + \ln (x^2+1)]\). This is exactly the same form as \(g(x)\).

Step 3 : Conclusion

From Steps 1 and 2, both graphical and algebraic methods confirm that \(f(x) = g(x)\). Hence, it is verified that \(f=g\) algebraically and graphically.

Key Concepts

Graphing Utility

Using a graphing utility is a fantastic way to visually compare mathematical functions. By plotting both functions, you can quickly observe their behavior and check for coincidences or differences. In this exercise, we use a graphing utility to plot the logarithmic functions \( f(x) = \ln \sqrt{x(x^2+1)} \) and \( g(x) = \frac{1}{2} [\ln x + \ln (x^2+1)] \). When both graphs are plotted in the same viewing window, they appear to coincide exactly, confirming that both functions are identical, at least graphically.

Graphing utilities help identify these coincidences without requiring complex algebraic manipulation at first glance. This initial graphical confirmation is valuable because it offers an intuitive understanding of the properties of functions before diving into algebraic proofs. Many graphing calculators and software programs available today make this process quick and accessible.

Here's why graphing utilities are beneficial:

• They provide a visual representation of functions, making abstract concepts more concrete.
• They offer a quick means of verification, saving time on manual calculations.
• They help in understanding the behavior of functions over a range of inputs, useful for detecting patterns or anomalies.

Logarithmic Functions

Logarithmic functions like \( f(x) = \ln \sqrt{x(x^2+1)} \) play a crucial role in mathematics due to their unique properties. The logarithm of a number is the power to which the base must be raised to produce that number. In this case, the base is \(e\), the natural logarithm.

A key feature of logarithms is their ability to transform multiplication into addition through properties such as:

• \( \ln ab = \ln a + \ln b \), known as the product rule.
• \( \ln a^n = n \ln a \), known as the power rule.

These properties enable us to simplify expressions, as seen in our exercise, where \( f(x) \) is re-expressed using these rules to match \( g(x) \).

Understanding these properties not only aids in solving the problem at hand but also opens up doors to solving more complex equations and real-world problems involving growth and decay processes, sound intensity levels, and the Richter scale, among others.

Algebraic Verification

Algebraic verification is a method used to prove that two equations are indeed equal by manipulating their algebraic expressions. In our case, this involves transforming \( f(x) = \ln \sqrt{x(x^2+1)} \) into the same form as \( g(x) = \frac{1}{2} [\ln x + \ln (x^2+1)] \). This is achieved through a series of steps using logarithmic properties.

First, \( f(x) \) is rewritten using the power rule as \( f(x) = \frac{1}{2} \ln(x(x^2+1)) \). Next, we apply the product rule, breaking down the expression further to \( \frac{1}{2}[\ln x + \ln (x^2+1)] \), which matches \( g(x) \).

Algebraic verification is crucial because:

• It provides definitive proof of equality between two expressions.
• It strengthens one's understanding of mathematical properties by applying them directly to solve problems.
• It builds confidence in using algebraic manipulation as a reliable method for solving equations.

This approach solidifies an intuitive grasp gained from the graphing utility, and confirms the accuracy of our observations through rigorous mathematical logic.