Problem 67 - Precalculus : Building Concepts and Connections

Question

In Exercises \(65-68,\) use a graphing utility with a decimal window. Graph \(f(x)=\ln e^{2} x\) and \(g(x)=\ln x\) on the same set of coordinate axes. Explain the relationship between the two graphs in terms of the properties of logarithms.

Step-by-Step Solution

Answer

The two graphs represent logarithmic functions. \(f(x) = \ln(e^{2}x)\) appears as a straight line with a slope of 2 due to the multiplication property of logarithms. \(g(x) = \ln(x)\) gradually increases for positive x and never crosses the x-axis, showing the characteristics of logarithmic functions.

Step 1: Plot the first function

The function \(f(x) = \ln(e^{2}x)\) simplifies to \(f(x) = 2x\) since the logarithm base \(e\) of \(e^{2}x\) is \(2x\). Plot this function on a graph.

Step 2: Plot the second function

The function \(g(x) = \ln(x)\) is the natural logarithm function. Plot this function on the same graph as the earlier function. Make sure to use the same scale.

Step 3: Analyze the shapes and properties of the graphs

Now compare the two graphs. Note that \(f(x) = 2x\) is a straight line passing through the origin with a slope of 2. Meanwhile, \(g(x) = \ln(x)\) is a curve that slowly increases for \(x > 0\). The logarithmic function never crosses the x-axis and it shoots down to negative infinity as x approaches 0.

Step 4: Explain the relationship using logarithmic properties

The relationship between these two functions is that they are both logarithmic in nature. The function \(f(x)\) is a specific case of the function \(g(x)\) where the input is multiplied by a factor of \(e^{2}\). The logarithmic function \(g(x) = \ln(x)\) stretches logarithmically as x increases where the other function \(f(x) = \ln(e^{2}x)\) is a straight line due to the multiplication property of logarithms.

Key Concepts

Graphing Logarithmic Functions

Understanding graphing logarithmic functions is a fundamental skill in mathematics that provides insight into the behavior of logarithms. When graphing a logarithmic function such as \( g(x) = \ln(x) \), you'll notice it has a unique curve: it approaches but never touches the y-axis, also known as an asymptote. For values of \( x > 0 \), the graph slowly rises, and as \( x \) gets closer to zero, the graph steepens and extends infinitely downwards. This behavior illustrates the key characteristic of logarithmic functions—they are undefined for non-positive values of \( x \).

Using a graphing utility helps students visualize these behaviors and see how different logarithmic functions compare to each other. For instance, plotting \( f(x) = \ln(e^{2}x) \), which simplifies to \( f(x) = 2x \), a linear function, on the same axes as \( g(x) \), highlights how the logarithmic function is transformed when its input is scaled. The linear nature of \( f(x) \), represented by a straight line through the origin with a slope of 2, contrasts heavily with the logarithmic curve of \( g(x) \), proving to be a visual representation of logarithmic properties in action.

Natural Logarithm

The natural logarithm is a specific type of logarithm that is denoted as \( \ln(x) \) and is based on the special number \( e \), approximately equal to 2.71828. This number \( e \) is the base of the natural logarithm, much like 10 is the common base for logarithms in the decimal system. The function \( g(x) = \ln(x) \) is the inverse of the exponential function \( e^x \) and has wide applications in science, engineering, and mathematics.

The natural logarithm has several important properties, one of which is that \( \ln(e) = 1 \) because the number \( e \) raised to the power of 1 is \( e \) itself. Additionally, \( \ln(1) = 0 \) because \( e \) to the power of 0 is 1. Graphing \( \ln(x) \) reveals its nature as a logarithmic function and its importance in describing growth processes and time decay in the natural world.

Logarithm Rules

Familiarity with logarithm rules is essential for understanding how to manipulate and simplify logarithmic expressions. These rules are mathematical properties that apply to all logarithms, regardless of base. Three primary rules often used are:

• Product Rule: \( \ln(ab) = \ln(a) + \ln(b) \), stating that the logarithm of a product is equal to the sum of the logarithms of the factors.
• Quotient Rule: \( \ln(a/b) = \ln(a) - \ln(b) \), meaning the logarithm of a quotient is the difference of the logarithms.
• Power Rule: \( \ln(a^b) = b \cdot \ln(a) \), showing that the logarithm of an exponent is the exponent multiplied by the log of the base.

In the exercise, the transformation of \( f(x) = \ln(e^{2}x) \) into \( f(x) = 2x \) actually uses the Power Rule, which simplifies the logarithm of \( e \) raised to an exponent, ultimately resulting in a linear function. Understanding and applying these rules enables one to tackle more complex logarithmic functions with ease.

Exponential Functions

Exponential functions represent the inverse process of taking a logarithm and are of the general form \( f(x) = a \cdot b^x \), where \( a \) is a constant, \( b \) is the base, and \( x \) is the exponent. These functions often describe situations where a quantity grows at a rate proportional to its current value, such as compound interest or population growth.

Graphically, exponential functions exhibit rapid growth or decay, which drastically differs from linear functions. The base of the exponential function determines whether the function is increasing or decreasing. If \( b > 1 \), the function grows exponentially, and if \( 0 < b < 1 \), it decays exponentially. To illustrate, the natural exponential function \( f(x) = e^x \), with \( e \) being the irrational base of the natural logarithm, is particularly important due to its unique mathematical properties and its appearance across various scientific disciplines.