Problem 51 - Contemporary Precalculus

Question

List the transformations that will change the graph of \(g(x)=\ln x\) into the graph of the given function. $$f(x)=2 \cdot \ln x$$

Step-by-Step Solution

Answer

Answer: A vertical stretch by a factor of 2.

Step 1: Identify the basic function

The basic function here is \(g(x) = \ln x\), which represents the natural logarithmic function with base e.

Step 2: Identify the given function

The given function \(f(x) = 2 \cdot \ln x\) is a transformation of the basic function \(g(x) = \ln x\).

Step 3: Compare the functions and find the transformation

Since \(f(x) = 2 \cdot \ln x = 2 \cdot g(x)\), we can identify that the transformation to go from \(g(x)\) to \(f(x)\) is a vertical stretch by a factor of 2.

Step 4: List the transformations

The transformations that will change the graph of \(g(x) = \ln x\) into the graph of the given function \(f(x) = 2 \cdot \ln x\) is a vertical stretch by a factor of 2.

Key Concepts

Graph Transformations

Transforming the graph of a function involves changing its appearance without altering its core characteristics. When we talk about graph transformations, we mean things like shifting, stretching, reflecting, and compressing a graph. In the exercise, the function we start with is the natural logarithmic function, noted as \( g(x) = \ln x \). The given problem asks for the transformation that maps \( g(x) \) to \( f(x) = 2 \cdot \ln x \). Here, the change involves scaling the graph up or down or possibly moving it along an axis.
To determine the specific transformation, we compare the two functions. We see that \( f(x) = 2 \cdot \ln x \) is derived from \(g(x)\) by applying a vertical stretch. The transformation affects all points on the graph by altering their vertical positions while their horizontal positions stay unchanged.
In summary, understanding graph transformations allows us to predict how a function graph will change when we apply different modifications to it, such as scaling for a vertical stretch.

Vertical Stretch

A vertical stretch happens when each point on a graph is moved away from the x-axis by multiplying the output by a factor. Think of it as pulling the graph upwards or downwards, making it taller or shorter. In the problem exercise, the transformation from \( g(x) = \ln x \) to \( f(x) = 2 \cdot \ln x \) is a classic example of a vertical stretch.
Here, since we multiply \( \ln x \) by 2, each y-value of the function becomes twice as large. This does not affect the x-values, which means the horizontal position of the points remains constant, only their vertical position increases.

• Example: If for \( g(x) = \ln(x) \), \( x = 1 \) produces \( \ln(x) = 0 \), for \( f(x) = 2 \cdot \ln(x) \), \( x = 1 \) still produces \( f(x) = 0 \), showing no vertical change at this point.
• Conversely, at \( x = e \) where \( \ln(e) = 1 \), for \( f(x) \), it becomes \( 2 \), showing the vertical doubling.

Keep in mind, a vertical stretch by a factor greater than 1 means each point will move further from the x-axis, leading to a taller appearance of the graph.

Natural Logarithms

Natural logarithms are a type of logarithm where the base is the constant \( e \). The value of \( e \) is approximately 2.71828, and it is an irrational number, meaning it can't be precisely written as a simple fraction. The function \( g(x) = \ln x \) is the natural logarithm function and represents the power to which we must raise \( e \) to obtain \( x \).
In mathematics, natural logs are widely used because they have unique properties that simplify many mathematical operations, especially involving growth patterns, like bacterial growth and compound interest.
The graph of the natural logarithm function \( g(x) = \ln x \) is logarithmic by nature. This means it increases quickly for values of \( x \) slightly greater than 1 but then increases more slowly as \( x \) becomes larger. This characteristic makes the natural logarithm useful for modeling and analyzing growth processes that start quickly and then slow. Thus, when transforming these types of functions, understanding the natural logarithm's behavior is crucial to seeing how certain transformations impact its graph.