Problem 68 - 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.$$\begin{array}{l}y_{1}=\ln \left(\frac{\sqrt{x}}{x+3}\right) \\\y_{2}=\frac{1}{2} \ln x-\ln (x+3)\end{array}$$.

Step-by-Step Solution

Answer

Based on the graph, table values and algebraic computation, it can be deduced that the two given equations \(y_{1}=\ln \left(\frac{\sqrt{x}}{x+3}\right)\) and \(y_{2}=\frac{1}{2} \ln x-\ln (x+3)\) are equivalent. This is because they generate the same values of 'y' for the same 'x' as seen in the graph and the table.

Step 1: Graph the equations using a graphing utility

Using a graphing tool (like Desmos, Geogebra, or a graphing calculator), input both logarithmic functions separately, \(y_{1}=\ln \left(\frac{\sqrt{x}}{x+3}\right)\) and \(y_{2}=\frac{1}{2} \ln x-\ln (x+3)\), to graph them. Adjust the viewing window to ensure that the key portions of both functions are clearly visible.

Step 2: Generate tables of values

Use the table feature in the graphing utility to generate a set of values for each equation. Note that these tables show the 'x' and 'y' values for each function. Be sure to select a range of 'x' values that includes significant points, such as where the two functions intersect.

Step 3: Interpret the graphs and tables

Examine the graphs and tables. They both provide information about how 'y' changes with 'x'. If the plots of the two functions overlap at any point, the 'y' values will be the same for the same 'x' value. This is a suggestion that the two equations may be equivalent.

Step 4: Algebraically verify the conclusion

From the logarithmic properties, it is known that \(\frac{1}{2} \ln x-\ln (x+3) = \ln \left(\frac{\sqrt{x}}{x+3}\right)\). Note that \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\) and \(\frac{1}{2} \ln x = \ln (\sqrt{x})\). This confirms that the two equations are indeed equivalent.

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools that help visualize mathematical functions, making it easier to comprehend their behavior. When tackling problems involving graphs, start by using a graphing utility like Desmos, Geogebra, or a graphing calculator. These tools allow you to input functions like \[ y_{1} = \ln \left(\frac{\sqrt{x}}{x+3}\right) \]and \[ y_{2} = \frac{1}{2} \ln x - \ln (x+3) \]and graph them in the same viewing window.

• Ensure that the graph shows all key features clearly by adjusting the settings.
• Look for intersections or overlapping portions on the graph, which may imply equality or similar behaviors between the equations.

Using the graphing utility's table feature can further enhance understanding by generating tables of specific values. This feature is invaluable as it helps pinpoint precise x-values where functions might intersect or overlap.

Algebraic Verification

Algebraic verification is a crucial counterpart to graphical analysis and involves proving that two expressions are equivalent using algebraic properties. For the given functions:\[ y_{1} = \ln \left(\frac{\sqrt{x}}{x+3}\right) \]and \[ y_{2} = \frac{1}{2} \ln x - \ln (x+3) \]It's essential to confirm the equivalency derived from graph interpretation through algebra. Here, we utilize properties of logarithms:

• Recall that \( \ln(a) - \ln(b) \) can be rewritten as \( \ln\left(\frac{a}{b}\right) \).
• Realize that \( \frac{1}{2} \ln x \) is equivalent to \( \ln(\sqrt{x}) \) due to the property \( n \ln(a) = \ln(a^n) \).

Combining these principles, we see:\[ \frac{1}{2} \ln x - \ln (x+3) = \ln (\sqrt{x}) - \ln (x+3) = \ln \left(\frac{\sqrt{x}}{x+3}\right) \]Algebraically verifying these transformations confirms the graphs' suggestion that the functions are equivalent.

Logarithmic Properties

Logarithmic properties are powerful tools that simplify complex expressions, making them easier to understand and apply. Some fundamental properties include:

• Product Property: \( \ln(a) + \ln(b) = \ln(ab) \).
• Quotient Property: \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \).
• Power Property: \( n \ln(a) = \ln(a^n) \).

In the problem, these properties help us understand that:- The expression \( \ln\left(\frac{\sqrt{x}}{x+3}\right) \) can be derived from the transformation \( \frac{1}{2} \ln x - \ln (x+3) \).Introducing these properties during algebraic verification processes ensures more accurate results and offers deeper insight into how logarithms transform expressions. Remembering these rules can simplify a variety of problems and make solving them faster and easier.