Problem 45 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Sketch the graph of the equation. $$y=\ln 2 x$$

Step-by-Step Solution

Answer

The graph of the equation \(y = \ln{2x}\) can be sketched by first identifying its domain as \(x > 0\) and range as \((-\infty, \infty)\). After calculating and plotting key points such as \((\frac{1}{2}, 0)\), \((1, \ln{2})\), and \((\frac{3}{2}, \ln{3})\), connect these points with a smooth curve. The curve starts from the x-axis at the point \((\frac{1}{2}, 0)\) and rises as x approaches infinity, while it tends toward negative infinity as x approaches 0. The curve never touches or crosses the y-axis.

Step 1: Determine Domain and Range

We need to first find the domain and range of \(y = \ln{2x}\). Remember that you can only take the logarithm of a positive number. Thus, we have: Domain: \(2x > 0\) \(x > 0\) The domain is \(x > 0\), which means the graph will only exist for all positive x values. To find the range, we should note that the graph of the natural logarithm will always output all real numbers (i.e., from \(-\infty\) to \(\infty\)) when its domain is positive: Range: \((-\infty, \infty)\)

Step 2: Identify Key Points

We will identify some key points on the graph to help us sketch the function. We can choose a few values of x within the domain and find their corresponding y values using the given equation. Let's try x values of \(x = \frac{1}{2}\), \(x = 1\), and \(x = \frac{3}{2}\): 1. For \(x = \frac{1}{2}\), \(y = \ln{(2 \cdot \frac{1}{2})} = \ln{1} = 0\) This gives us the point \((\frac{1}{2}, 0)\). 2. For \(x = 1\), \(y = \ln{(2 \cdot 1)} = \ln{2}\) This gives us the point \((1, \ln{2})\). 3. For \(x = \frac{3}{2}\), \(y = \ln{(2 \cdot \frac{3}{2})} = \ln{3}\) This gives us the point \((\frac{3}{2}, \ln{3})\).

Step 3: Sketch the Graph

Now that we have the domain, range, and key points, we can sketch the graph of \(y = \ln{2x}\): 1. Start by drawing the x-axis (horizontal) and y-axis (vertical). 2. Add tick marks at \(x = 1\), \(x = \frac{1}{2}\), and \(x = \frac{3}{2}\) on the x-axis. 3. Plot the key points: \((\frac{1}{2}, 0)\), \((1, \ln{2})\), and \((\frac{3}{2}, \ln{3})\). 4. Draw a smooth curve connecting the points that starts from the x-axis at the point \((\frac{1}{2}, 0)\) and goes through the other points. As x approaches infinity, the curve will continue to rise, and as x approaches 0, the curve will tend toward negative infinity. 5. Remember that the curve will not touch or cross the y-axis since the natural logarithm is undefined for a non-positive x. By following these steps, you should now have a rough sketch of the graph of \(y = \ln{2x}\).

Key Concepts

Domain and Range

When sketching the graph of a function, understanding the domain and range is crucial. For the function \(y = \ln{2x}\), the domain refers to the set of all \(x\) values for which the function is defined. Since the natural logarithm only accepts positive values, we need \(2x > 0\), leading to \(x > 0\). Thus, the domain is all positive real numbers, or \(x > 0\).

The range, on the other hand, describes all possible \(y\) values the function can take. For \(y = \ln{2x}\), no matter what positive values \(x\) takes, the resulting \(y\) can span any real number. This means the range of \(y = \ln{2x}\) is all real numbers, or \((-\infty, \infty)\).

Natural Logarithm

The natural logarithm function, represented as \(\ln(x)\), is a fundamental concept in mathematics that deals with the logarithm to the base \(e\), where \(e\) is approximately 2.718. For example, \(\ln{e} = 1\) because \(e^1 = e\). It's important to understand that the natural logarithm is largely used in calculus for its unique connection to growth rates.

With the function \(y = \ln{2x}\), we modify the basic \(\ln(x)\) by introducing a multiplier to the argument inside the logarithm. This adjusts the graph such that it's stretched horizontally. The primary characteristic of a natural logarithm graph is it always contains a vertical asymptote, meaning it skyrockets to infinity as the \(x\) approaches zero, allowing it only to accept positive input values.

Key Points

Identifying key points on a graph is a helpful tool in sketching the curve accurately. For the equation \(y = \ln{2x}\), we start by evaluating it at different \(x\) values that fall within its domain.

• When \(x = \frac{1}{2}\), \(y = \ln{(2 \cdot \frac{1}{2})} = \ln{1} = 0\).
• When \(x = 1\), \(y = \ln{(2 \cdot 1)} = \ln{2}\).
• When \(x = \frac{3}{2}\), \(y = \ln{(2 \cdot \frac{3}{2})} = \ln{3}\).

These points, \((\frac{1}{2}, 0)\), \((1, \ln{2})\), and \((\frac{3}{2}, \ln{3})\), help us plot essential positions on the graph. By connecting these points smoothly, we can visualize how the natural logarithm function progresses as \(x\) increases.

Smooth Curve Drawing

Drawing a smooth curve is the art of connecting plotted points to form the graph of a function. For \(y = \ln{2x}\), it involves carefully linking points on the graph to reflect the continuous nature of the logarithm function.

Start by plotting the identified key points on a graph with the x-axis and y-axis drawn. Once the points are plotted:

• Draw a curve that starts at the point \((\frac{1}{2}, 0)\), which lies on the x-axis.
• The curve should extend through the other identified points \((1, \ln{2})\) and \((\frac{3}{2}, \ln{3})\).
• As \(x\) approaches zero, ensure that the curve trends downward toward negative infinity, reflecting the behavior of the natural logarithm near zero.
• Ensure the curve does not touch or cross the y-axis, as \(\ln{2x}\) is undefined for non-positive \(x\).

By drawing the function smoothly, we capture the natural flow and behavior of the natural logarithm, illustrating its increase as \(x\) becomes larger.