Problem 66 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) What do the graphs and tables suggest? Verify your conclusion algebraically.$$\begin{array}{l}y_{1}=\ln 9 x^{3} \\\y_{2}=\ln 9+3 \ln x\end{array}$$

Step-by-Step Solution

Answer

Both the graphical and table observations, and the algebraic verification confirm that the two equations are identical, meaning \(y_{1} = y_{2}\) for all valid \(x\).

Step 1: Graph the equations

First, enter each equation into the function plotter of a graphing utility, keeping the viewing windows consistent for both to accurately compare them. Observe the shapes of the graphs and how they relate to each other.

Step 2: Create a table of values

Next, use the table feature of the graphing utility to generate a table of values for \(x\), \(y_{1}\), and \(y_{2}\). Look for patterns in the values. Note the outputs for each equation whenever \(x\) is the same.

Step 3: Interpret the graphs and tables

Pay attention to whether the graphs overlap or intersect, look at the points where they do so. Compare the corresponding \(y_{1}\) and \(y_{2}\) values in the table for the same \(x\).If they are equal or very close in value then suggests that \(y_{1}=y_{2}\)

Step 4: Algebraic verification

The observed patterns from the graphs and tables suggest that \(y_{1} = y_{2}\). To verify this algebraically, set the two expressions equal to each other and simplify: \( \ln 9x^{3} = \ln 9 + 3 \ln x\). Using properties of logarithms, where \( \log(a)+\log(b)=\log(ab)\) and \( n \log(a)=\log(a^n) \), so\[ \ln 9x^{3} = \ln 9+ \ln x^{3}\], which simplifies to \( \ln 9x^{3} = \ln 9x^{3}\). Thus, this confirms algebraically that \(y_{1} = y_{2}\).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential for simplifying logarithmic expressions and solving logarithmic equations. Logarithms, which are the inverse functions of exponentials, have unique properties that allow us to manipulate them in various ways.

One fundamental property is the Product Rule, which states that the logarithm of a product is equal to the sum of the logarithms of the individual factors: \[ \ln(ab) = \ln(a) + \ln(b) \]. Another key property is the Quotient Rule, which shows how to handle division: \[ \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \]. Lastly, the Power Rule allows us to deal with exponents: \[ \ln(a^n) = n\ln(a) \].

In the given exercise, these properties help us verify algebraically that the two logarithmic expressions are equivalent. The relationship between the logarithmic properties and the algebraic expression becomes evident when we understand how to apply these rules to combine and break apart logarithmic terms.

Logarithmic Equations

Logarithmic equations involve variables within the logarithm and require a strong grasp of logarithmic properties to solve. To solve a logarithmic equation, we often start by using properties of logarithms to combine or separate the logarithmic terms. In some cases, we transform the logarithmic equation into an exponential form to solve for the variable.

For example, in our exercise, we have the equation \( \ln 9x^{3} = \ln 9 + 3 \ln x \). By applying logarithmic properties, we simplify this to show that \( \ln 9x^{3} = \ln 9x^{3} \)—hence, verifying the equality of the two functions. The ability to manipulate and simplify using the properties of logarithms key to solving logarithmic equations, as illustrated in the exercise.

Graphing Utility Use in Algebra

The utilization of graphing utilities is a powerful method for understanding and solving algebraic problems, particularly when working with functions, such as logarithmic functions. Graphing utilities enable us to visually compare functions and comprehend their behavior without solely relying on algebraic manipulation.

When graphing the logarithmic functions \( y_1 = \ln 9 x^{3} \) and \( y_2 = \ln 9 + 3 \ln x \) with a graphing utility, we observe their shapes and intersections. This visual comparison, along with the tables of values that these utilities can generate, provides an intuitive understanding of how the functions relate to each other. As we saw in the exercise, graphing both functions helps us hypothesize that they are equivalent. Subsequent algebraic verification aligns with the visual insights provided by the graphs and tables, affirming the utility of graphing tools in algebra to confirm findings.