Problem 67 - Calculus for Biology and Medicine

Question

Use a logarithmic transformation to find a linear relationship between the given quantities and determine whether a log-log or log-linear plot should be used to graph the resulting linear relationship. $$ f(x)=3 x^{1} $$

Step-by-Step Solution

Answer

Use a log-log plot for the function; it linearizes the expression \( f(x) = 3x^1 \).

Step 1: Understand Logarithmic Transformation Types

A logarithmic transformation can either take both variables to a log scale (log-log) or transform just the dependent variable (log-linear). To figure out which transformation works, we first know that a log-log plot is used when both variables have power relationship and a log-linear plot when only one side has an exponential relationship.

Step 2: Analyze the Given Function

The function given is \( f(x) = 3x^1 \). Here, \( x^1 \) is a power function with exponent 1. It implies a linear relationship between \( f(x) \) and \( x \). For a general expression in terms of \( y = ax^b \), a log-log transformation could facilitate viewing the linear relationship when plotted.

Step 3: Apply Logarithmic Transformation

Apply the logarithmic transformation on the function. We take the natural log (or common log) of both sides:\[\ln(f(x)) = \ln(3x^1)\]This can be expanded using logarithmic properties:\[\ln(f(x)) = \ln(3) + 1 \cdot \ln(x)\]

Step 4: Recognize the Linear Equation Format

The resulting equation \( \ln(f(x)) = \ln(3) + 1 \cdot \ln(x) \) is in the format \( Y = A + B \cdot X \), with \( Y = \ln(f(x)) \), \( A = \ln(3) \), \( B = 1 \), and \( X = \ln(x) \). This linear format indicates a log-log plot is appropriate.

Step 5: Determine the Plot Type

From the linear equation derived, it is clear that both the dependent variable \( f(x) \) and independent variable \( x \) should be plotted on a logarithmic scale, indicating a log-log plot. This confirms the linear relationship between the two transformed variables.

Key Concepts

Linear Relationship

A linear relationship describes a direct connection between two quantities. This means if one quantity changes, the other changes in a consistent manner, maintaining a constant ratio. This relationship is often represented graphically as a straight line. Understanding linear relationships is crucial:

• A linear equation can be expressed as: \( y = mx + c \), where \( m \) is the slope, indicating the steepness of the line.
• In the context of logarithmic transformations, when you can transform a nonlinear equation into this linear format, it implies a simple, direct relationship between your variables.
• This makes predicting the behavior of one variable easier based on the other.

In our exercise, the original function \( f(x) = 3x^1 \) was transformed into a linear equation: \( \ln(f(x)) = \ln(3) + 1 \cdot \ln(x) \). Here, the relationship is linear in the transformed scale, as evident from the equation’s format similar to \( Y = A + B \cdot X \). This reveals a clear linear connection after transformation.

Log-Log Plot

A log-log plot is a type of graph used when both variables involved are better understood on a logarithmic scale. This often applies when dealing with power functions, as transforming them can help reveal a linear relationship.

• When both axes of a plot are on a log scale, changes in one quantity relate to proportional changes in another.
• This is particularly powerful because certain complex relationships in their raw form appear straighter and more analyzable after transformation.
• The log-log plot is most useful when examining relations like power laws.

In our scenario, the function \( f(x) = 3x^1 \) was transformed with a log-log approach. Both \( f(x) \) and \( x \) were plotted on logarithmic scales resulting in the linear appearance of the equation \( \ln(f(x)) = \ln(3) + 1 \cdot \ln(x) \). This aligns perfectly with a log-log visualization, making it easier to work with and interpret.

Log-Linear Plot

A log-linear plot features one axis (usually the dependent variable) on a logarithmic scale. This type of plot is useful when the relationship between the two variables involves an exponential behavior. However, it is different from a log-log plot which uses logs on both axes.

• In a log-linear plot, you might plot \( Y = a \cdot b^x \) by transforming the equation to \( \ln(Y) = \ln(a) + x \cdot \ln(b) \). Here, the Y-axis is on a log scale, highlighting linear trends in exponential relationships.
• This transformation allows exponential growth or decay to appear as a straight line, facilitating easier analysis.

In the exercise, a log-linear transformation was not necessary because the function \( f(x) = 3x^1 \) depicts a power function, rather than an exponential one. Thus, a log-log plot was more appropriate to reveal the linear character of the relationship between \( f(x) \) and \( x \).

Power Function

A power function is a type of mathematical expression where one variable is raised to a fixed power. It has the form \( y = ax^b \), where \( a \) and \( b \) are constants and \( x \) is the variable. Power functions often appear in many natural phenomena.

• The relationship between \( y \) and \( x \) is nonlinear, depending directly on the power \( b \) — the exponent.
• They can take various forms, such as squared, cubic, or higher powers, influencing the shape and characteristics of their graphs.

A key feature of power functions is that they are perfect candidates for log-log transformations. By taking logarithms of both sides, you transform them into linear functions, making them easier to plot and analyze. In the provided exercise, \( f(x) = 3x^1 \) already has a straightforward linear form as a power function because \( b = 1 \). Transforming this with logarithms gave \( \ln(f(x)) = \ln(3) + 1 \cdot \ln(x) \), a clear linear equation ideal for both visualization and interpretation. This demonstrates one of the major advantages of using logarithmic transformations on power functions.