Problem 69 - Elementary and Intermediate Algebra

Question

Use a graphing calculator to graph each function. See Objective 2. See Using Your Calculator: Graph Base-e Logarithmic Functions. $$f(x)=\ln \left(\frac{1}{2} x\right)$$

Step-by-Step Solution

Answer

Graph \( f(x) = \ln\left(\frac{1}{2}x\right) \) using a calculator to see a horizontally stretched \( \ln(x) \) curve.

Step 1: Identify the Function

The function given is \( f(x) = \ln\left(\frac{1}{2}x\right) \). This is a natural logarithmic function where \( x \) is multiplied by \( \frac{1}{2} \).

Step 2: Understand the Graphing Aspect

To graph \( f(x) = \ln\left(\frac{1}{2}x\right) \), recognize that this is a transformation of the basic natural logarithm function \( \ln(x) \). The \( \frac{1}{2} \) factor stretches the graph horizontally by a factor of 2.

Step 3: Input Function in Calculator

Enter the function \( f(x) = \ln\left(\frac{1}{2}x\right) \) into your graphing calculator. Use the calculator's natural log function (often labeled as "ln") and enter the expression correctly, ensuring the fraction \( \frac{1}{2} \) is inside the logarithm.

Step 4: Set Appropriate Viewing Window

Choose an appropriate window to view the graph. Since \( \ln(x) \) is undefined for \( x \leq 0 \), ensure the x-values in the window are positive. A suggested range could be \( x: (0, 10) \) and \( y: (-5, 5) \).

Step 5: Observe and Interpret the Graph

After graphing, observe the curve that resembles \( \ln(x) \) but stretched horizontally. The graph should cross the x-axis at \( x = 2 \) because \( \ln\left(\frac{1}{2}(2)\right) = \ln(1) = 0 \).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a fundamental logarithmic function that uses the constant "e" (approximately 2.718) as its base. It's commonly used in calculus and real-world applications due to its natural properties in growth processes. When we talk about \( \ln(x) \), we are finding the power to which the base "e" must be raised to obtain "x." For example, if \( \ln(x) = 1 \), it means that \( e \) raised to the power of 1 is equal to "x," which makes \( x = e \).

Natural logarithm functions have several key properties:
- They are undefined for \( x \leq 0 \), which means they only take positive real numbers as inputs. This is why the graph of \( \ln(x)\) only exists on the right side of the y-axis.
- The graph of \( \ln(x) \) is a continuous curve that rises slowly as "x" increases and heads toward negative infinity as "x" approaches zero from the positive side.

Transformation of Graphs

Transformation of graphs involves changing a graph's position, shape, or size. For example, stretches and shifts are basic transformations applied to graph functions. In our given function \( f(x) = \ln \left(\frac{1}{2}x\right) \), we are dealing with a horizontal stretch.

Here are the transformational steps for \( \ln \left(\frac{1}{2}x\right) \):

• The factor \( \frac{1}{2} \) impacts the graph of \( \ln(x) \) by stretching it horizontally by a factor of 2. This means that every "x" value on the standard \( \ln(x) \) graph will be "stretched," appearing twice as far from the y-axis.
• Horizontal stretches, like the one prompted by \( \frac{1}{2} \), change where the graph crosses specific points on the x-axis. In this case, instead of crossing at \( x = 1 \) like a standard \( \ln(x)\) graph, it crosses at \( x = 2 \), due to the transformation\( \frac{1}{2}x = 1 \) when \( x = 2 \).

Understanding these transformations helps you predict the shape and position of graphs without explicitly plotting many points.

Using Graphing Calculators

Graphing calculators are powerful tools that allow you to visualize complex equations like logarithms easily. When graphing functions such as \( f(x) = \ln \left(\frac{1}{2}x\right) \), the graphing calculator can produce accurate visual representations which are invaluable for comprehension.

Here's a straightforward process to use on a graphing calculator:

• First, access the natural log function often labeled "ln" on the calculator. Ensure you enter the expression \( \frac{1}{2}x \) correctly within the parentheses.
• Next, set an appropriate graph window. Since natural logarithms are undefined at non-positive values, choose a range like \( x: (0, 10) \) to view the relevant portions of the graph.
• Lastly, interpret the results on the screen. The calculator should display a curve similar to \( \ln(x) \), horizontally stretched, which crosses the x-axis at \( x = 2 \). Observing how the curve behaves as it stretches can aid in deepening your understanding of transformations.

Using graphing calculators effectively can simplify your interaction with algebraic expressions, especially those involving logarithms.