Problem 87 - Precalculus

Question

Use a graphing calculator to plot \(y=\ln \left(x^{2}\right)\) and \(y=2 \ln x\) Are they the same graph?

Step-by-Step Solution

Answer

Yes, they are the same graph for \(x > 0\).

Step 1: Understand the Functions

We have two functions: \(y = \ln(x^2)\) and \(y = 2\ln(x)\). Although they may look different at first glance, notice that they might be transformations of each other. They both involve logarithms and powers of \(x\).

Step 2: Simplify Each Function

Let's rewrite the first function using the power rule of logarithms: \(\ln(x^2) = 2\ln(x)\). This tells us that the expression \(\ln(x^2)\) can be simplified to the same form as \(2\ln(x)\).

Step 3: Compare the Functions

Now that we have \(y = 2\ln(x)\) for both functions, we can see they are actually the same. Thus, both original functions simplify to \(y = 2\ln(x)\), showing they represent the same graph.

Step 4: Consider Domain Restrictions

Both functions \(y = \ln(x^2)\) and \(y = 2\ln(x)\), after simplification, depend on the domain of the logarithm function. However, \(\ln(x^2)\) takes positive values for all \(x\) except zero, whereas \(2\ln(x)\) only takes positive values for \(x > 0\). This difference doesn't affect their graph shape as both become undefined at \(x \leq 0\).

Step 5: Graph the Functions

Graph \(y = \ln(x^2)\) and \(y = 2\ln(x)\) using a graphing calculator. You will observe their graphs overlap completely for \(x > 0\), confirming they are indeed the same graph despite apparent differences in form.

Key Concepts

Graph Transformations

Graph transformations play a significant role when dealing with logarithmic functions, as they help us visualize and understand how changes in function expressions affect their overall graph. For instance, the functions given in the exercise seem distinct at first glance:

• \(y=\ln(x^2)\)
• \(y=2\ln(x)\)

However, using the power rule for logarithms, we discover they are algebraically the same. Transformations can include shifts, stretches, compressions, and reflections. In this context, simplifying \(\ln(x^2)\) into \(2\ln(x)\) is akin to recognizing a vertical stretch by a factor of 2. Although this transformation might not change the overall graph shape drastically, it demonstrates the importance of simplification techniques. These transformations can be applied to better predict and confirm graph overlaps or equivalencies of different expressions.

Domain of a Function

Understanding the domain of logarithmic functions is crucial when graphing them, especially since these functions are not defined for all values of \(x\). Let's break down the two functions we're dealing with:

• For \(y = \ln(x^2)\), the domain is all real numbers except zero because \(x^2\) is always positive except when \(x = 0\).
• For \(y = 2\ln(x)\), the domain is restricted to \(x > 0\) as the logarithm is not defined for zero or negative values.

While both expressions simplify to \(y = 2\ln(x)\) algebraically, it's critical to note that \(y = \ln(x^2)\) could theoretically allow for negative \(x\) if we only consider the expression itself without domain restrictions. The actual graph, however, ignores these extraneous possibilities, focusing strictly on positive values due to the properties of logarithms. When graphing these expressions, it's essential to ensure that domain restrictions are duly considered to avoid misleading visual interpretations.

Graphing Calculator

In this exercise, a graphing calculator proves quite handy in visually verifying whether the two logarithmic functions are indeed similar, despite their initial algebraic differences. By inputting both \(y = \ln(x^2)\) and \(y = 2\ln(x)\) into the calculator, we can observe their graphical representations. Here are some tips for using a graphing calculator effectively:

• Ensure both functions are correctly entered into the calculator to avoid errors.
• Set an appropriate window for your graph to visualize the overlap for \(x > 0\), enhancing comparison accuracy.
• Watch out for domain errors that the calculator might indicate when inputs fall outside the defined domain.

Ultimately, the graph should confirm that both functions are equivalent, as their lines will overlap perfectly for all positive \(x\). This visual aid not only validates the algebraic simplification but also reinforces the importance of understanding function domains during graphing exercises.