Problem 87 - 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.$$y_{1}=\ln x+\frac{1}{2} \ln (x+1), \quad y_{2}=\ln (x \sqrt{x+1})$$

Step-by-Step Solution

Answer

The two equations \(y_{1}=\ln x+\frac{1}{2} \ln (x+1)\) and \(y_{2}=\ln(x\sqrt{x+1})\) are equal to each other. Analyzing the corresponding graphs and tables of these equations indicates that as these two graphs coincide with the same y-values for every x-value in the table. The algebraic verification supports this result, as simplifying \(y_{1}\) yields the same logarithmic expression as \(y_{2}\).

Step 1: Graph generation

Use a graphing utility to plot the given equations, \(y_{1}\) and \(y_{2}\), in the same viewing window to visually assess their relationship. Focus on similarities or differences in the shape and position of the two graphs.

Step 2: Creating data tables

Use the table feature of the graphing utility to create a table of values for each equation. Compare these values to further understand any patterns or discrepancies between the two equations.

Step 3: Drawing conclusions from graphs and tables

By closely inspecting the graphs and tables, decide what they suggest about the relationship between \(y_{1}\) and \(y_{2}\). Do the graphs coincide? Do the y-values in the tables correspond for every x-value?

Step 4: Algebraic verification

Verify the conclusion algebraically by transforming both equations into a comparable form, if possible. By applying the rule \(ln(a) + ln(b) = ln(ab)\) in the \(y_1\) equation, it can be rewritten as \(y_1 = ln(x(x+1)^{1/2})\). These two equations are equal to each other because the expression inside the logarithm is the same for both.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool that allows you to visualize mathematical functions. In this exercise, you will use a graphing utility to plot the logarithmic functions defined by equations

• \( y_1 = \ln x + \frac{1}{2} \ln(x+1) \)
• \( y_2 = \ln(x \sqrt{x+1}) \)

Seeing these graphs on the same viewing window helps you understand their relationship.
Graphing utilities can help you notice key characteristics like intersections, slopes, or asymptotic behaviors. This is especially helpful with logarithmic functions, which can often be complex. By observing these characteristics, you can make initial guesses about how these functions relate to one another.

Table Feature

The table feature in a graphing utility allows you to generate a list of values for a function's input and corresponding output. This can give precise data points to support or refine your observations from the graph. For the functions

• \( y_1 = \ln x + \frac{1}{2} \ln(x+1) \)
• \( y_2 = \ln(x \sqrt{x+1}) \)

You can plug in various x-values to see how the y-values for both functions compare. By examining these tables, look for points where the values of \( y_1 \) and \( y_2 \) are equal or note any discrepancies. Consistencies across many points can indicate that the graphs reflect the same function, while inconsistencies might suggest differences.

Algebraic Verification

Upon observing both the graph and table, you might suspect that the equations coincide. To be certain, algebraic verification is necessary. By using logarithmic properties, try to transform the first equation so that it resembles the second.
In this case, apply the property \( \ln(a) + \ln(b) = \ln(ab) \) to \( y_1 = \ln x + \frac{1}{2} \ln(x+1) \). It becomes \( y_1 = \ln(x(x+1)^{1/2}) \), which is identical to \( y_2 = \ln(x \sqrt{x+1}) \).
This algebraic manipulation confirms the observation from the graph and table, showing they are indeed the same function. Algebraic verification is an important step in many math problems to turn a graphical guess into a proof.

Logarithmic Properties

Understanding logarithmic properties is essential when dealing with these types of functions. Here are some key properties:

• The product property: \( \ln(a) + \ln(b) = \ln(ab) \)
• The power property: \( \ln(a^b) = b\ln(a) \)
• The quotient property: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)

These properties allow you to manipulate and simplify logarithmic functions. In the context of this exercise, using the product and power properties transformed the original expression of \( y_1 \) into a form that matches \( y_2 \). This transformation helps identify that two seemingly different expressions are actually the same, which is crucial for confirming function equivalence.