Problem 35 - 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. $$ f(x)=\ln \frac{x^{2}}{4}, \quad x>0, \quad g(x)=2 \ln x-\ln 4 $$

Step-by-Step Solution

Answer

Both graphically and algebraically, the function \(f(x) = \ln\left(\frac{x^2}{4}\right)\) is verified to be equal to the function \(g(x) = 2\ln(x) - \ln(4)\).

Step 1: Graphical Verification

Use a graphing utility to plot the functions \(f(x) = \ln\left(\frac{x^2}{4}\right)\) and \(g(x) = 2\ln(x) - \ln(4)\) in the same viewing window.\n\nIf both functions produce the same line on the graph, then \(f(x)\) equals \(g(x)\) graphically.

Step 2: Algebraic Verification

Algebraically verify \(f(x) = g(x)\) by simplifying both functions until they match.\n\nFor the function \(f(x)\), use the property of logarithms that states \(\ln\left(\frac{a}{b}\right)=\ln(a)-\ln(b)\) to rewrite \(f(x)\) as \(f(x)= \ln(x^2) - \ln(4)\).\n\nFor the function \(g(x)\), it is already in a simplified form, \(g(x)= 2\ln(x) - \ln(4)\).\n\nBecause \(2\ln(x)\) is equal to \(\ln(x^2)\) according to the properties of logarithms, \(g(x)\) can be rewritten as \(g(x)=\ln(x^2)-\ln(4)\). \n\nNow, \(f(x)\) and \(g(x)\) are the same, which validates the equality \(f(x)=g(x)\) algebraically.

Key Concepts

Graphing Utility

One of the most visual methods for verifying the equality of two functions is the use of a graphing utility. A graphing utility, which can be software or a dedicated device, provides a graphical representation of functions to help us understand their behavior visually. In this exercise, we are tasked with plotting the functions:

• \(f(x) = \ln\left(\frac{x^2}{4}\right)\)
• \(g(x) = 2\ln(x) - \ln(4)\)

To determine if \(f(x)\) and \(g(x)\) are equal, plot both on the same graph. If their graphs perfectly overlap, it suggests that the two functions are equivalent for the values of \(x\) in the domain. This visual method offers an immediate, albeit not entirely precise, way to verify the equality.
However, the graphical approach has limitations. It can show that two functions appear to be the same for a specific range of \(x\)-values but not confirm it for all possible values. This is why combining graphical methods with algebraic verification offers more confidence in your results.

Algebraic Verification

Once we've suggested equality using a graphing utility, a more precise algebraic verification is required to confirm it robustly. Algebraic verification relies on mathematical rules and properties to simplify and compare functions.
For this exercise, start with function \(f(x) = \ln\left(\frac{x^2}{4}\right)\). Apply the logarithm rule: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\). Using this, rewrite \(f(x)\) as:

• \(f(x) = \ln(x^2) - \ln(4)\)

For \(g(x) = 2\ln(x) - \ln(4)\), no further simplification is needed as it's already in a form we can use.
To continue, apply another logarithmic property: \(2\ln(x) = \ln(x^2)\). This allows us to rewrite \(g(x)\) as:

• \(g(x) = \ln(x^2) - \ln(4)\)

Now we see that both functions simplify to the same expression, which confirms that \(f(x) = g(x)\). This algebraic approach provides a rigorous check that the functions are indeed equal across all \(x>0\).

Logarithm Rules

Understanding logarithm rules is essential for manipulating and verifying equations involving logarithms. Here are some key rules you'll find useful:

• **Product Rule**: \(\ln(ab) = \ln(a) + \ln(b)\)
• **Quotient Rule**: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)
• **Power Rule**: \(\ln(a^c) = c\ln(a)\)

In our exercise, these rules were vital in showing that two seemingly different functions are, in fact, equivalent. By recognizing when and how to apply these rules, you unlock the capability to simplify complex logarithmic expressions.
As demonstrated:- The quotient rule was used to break down \(\ln\left(\frac{x^2}{4}\right)\) into a difference of logarithms.- The power rule allowed us to equate \(2\ln(x)\) with \(\ln(x^2)\), simplifying the expression further.
Mastering these rules enables you to confidently transform and prove logarithmic expressions, reinforcing your algebraic skills.