Problem 49 - Calculus for Biology and Medicine

Question

use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-linear plot. $$ y=2 e^{-1.2 x} $$

Step-by-Step Solution

Answer

The linear relationship is \( \ln y = -1.2x + \ln 2 \) and plots as a straight line with slope -1.2 on a log-linear plot.

Step 1: Understand the Relationship

The given equation is an exponential function: \( y = 2 e^{-1.2x} \). Our goal is to transform this into a linear relationship using logarithms.

Step 2: Apply Logarithmic Transformation

Take the natural logarithm of both sides: \( \ln y = \ln(2 e^{-1.2x}) \). According to the logarithm product property, this simplifies to \( \ln y = \ln 2 + \ln e^{-1.2x} \).

Step 3: Simplify Using Logarithm Properties

Know that \( \ln e^{-1.2x} = -1.2x \) (since the natural logarithm and exponential are inverse functions). The equation becomes \( \ln y = \ln 2 - 1.2x \).

Step 4: Identify Linear Equation Format

The resulting equation \( \ln y = -1.2x + \ln 2 \) is in the form \( y = mx + c \), where \( m = -1.2 \) and \( c = \ln 2 \). This indicates a linear relationship on a log-linear plot.

Step 5: Graph the Transformation

Plot \( \ln y \) versus \( x \) on a graph. The slope of the line is \(-1.2\), and it intercepts the \( \ln y \)-axis at \( \ln 2 \). Use a log-linear plot, where the \( y \)-axis is logarithmic, to represent this linear relationship.

Key Concepts

Log-Linear Plot

A log-linear plot is a type of graph that serves as an effective tool for visualizing data that grows exponentially. In such plots, the horizontal axis is linear, while the vertical axis is logarithmic. This means that instead of counting units equally as in a regular graph, the log scale compresses higher values relative to lower ones. As a result, exponential relationships convert into straight lines.

• By plotting the natural logarithm of the dependent variable against the independent variable, any exponential growth or decay in the data presents itself as a linear trend.
• This makes it much easier to interpret empirical results and understand underlying relationships.

In the exercise, the log-linear plot facilitated the transformation of the exponential equation into a linear form. By plotting \( \ln y \) against \( x \), a clear straight line with slope \(-1.2\) and intercept at \( \ln 2 \) is achieved. This visualization aids in simplifying complex dynamics into more manageable knowledge.

Linear Relationship

In mathematics, a linear relationship refers to a kind of direct association between two variables. Such a relationship is characterized by a straight line when graphed on a Cartesian coordinate system. Its general form is expressed by the equation \( y = mx + c \), where:

• \( y \) represents the dependent variable.
• \( x \) stands for the independent variable.
• \( m \) is the slope of the line, indicating the rate of change of \( y \) with respect to \( x \).
• \( c \) is the y-intercept, where the line crosses the y-axis.

The exercise illustrated how taking the natural logarithm of the exponential function transformed it into a linear equation: \( \ln y = -1.2x + \ln 2 \). This step highlighted the inverse relationship between exponential growth and linear transformations, allowing easier interpretation of the data's behavior.

Exponential Function

An exponential function describes situations where a quantity grows or decays at a rate proportional to its current value. It is typically represented by the formula \( y = a e^{bx} \), where:

• \( y \) is the output or dependent variable.
• \( a \) stands for the initial value, starting point, or amplitude.
• \( b \) represents the rate of growth (positive) or decay (negative).
• \( e \) is the base of natural logarithms, approximately 2.718.
• \( x \) is the input or independent variable.

In the original exercise, we dealt with \( y = 2 e^{-1.2x} \), showcasing exponential decay since the exponent \(-1.2\) is negative. This means that as \( x \) increases, \( y \) decreases rapidly at first but then slows down due to the nature of the exponential function. Through logarithmic transformation, this complex behavior was simplified into a linear form, demonstrating the utility of mathematical manipulations in revealing fundamental patterns.