Problem 49 - Calculus for Biology and Medicine

Question

Use a logarithmic transformation to find \(a\) linear relationship between the given quantities and graph the \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l i n e a r ~ p l o t . ~ $$ v=2 e^{-1.2 x} $$

Step-by-Step Solution

Answer

The linear relationship is \( \ln(v) = -1.2x + \ln(2) \). Plot \( \ln(v) \) against \( x \) for a straight line.

Step 1: Understand the Problem

We need to find a linear relationship from the given exponential function \( v = 2e^{-1.2x} \). This involves transforming the equation using logarithms.

Step 2: Apply Logarithmic Transformation

Take the natural logarithm on both sides of the equation to facilitate linearization: \( \ln(v) = \ln(2e^{-1.2x}) \). Simplify this using the property of logarithms: \( \ln(v) = \ln(2) + \ln(e^{-1.2x}) \).

Step 3: Simplify the Logarithmic Expression

Since \( \ln(e^{-1.2x}) = -1.2x \), you can rewrite the equation as: \( \ln(v) = \ln(2) - 1.2x \). This is a linear equation in the form \( y = mx + c \), where \( y = \ln(v) \), \( m = -1.2 \), and \( c = \ln(2) \).

Step 4: Interpret the Linearized Equation

Identify the transformed variables: \( y = \ln(v) \) and \( x \) remains as the independent variable. The equation \( \ln(v) = \ln(2) - 1.2x \) represents a line with slope \(-1.2\) and y-intercept \(\ln(2)\).

Step 5: Plot on a Log-Linear Graph

Plot \( y = \ln(v) \) against \( x \) on a graph. Since it's a linear plot, use the linearized equation \( y = -1.2x + \ln(2) \). The slope of the line is \(-1.2\) and it intercepts the y-axis at \( \ln(2) \).

Key Concepts

Linear Relationship

In mathematics, a linear relationship is represented by a straight line on a graph. This relationship describes how one quantity changes at a consistent rate with respect to another. When examining a linear relationship, you often look for the equation of a line in the form \( y = mx + b \), where:

• \( y \) is the dependent variable.
• \( m \) is the slope of the line, indicating the rate of change.
• \( x \) is the independent variable.
• \( b \) is the y-intercept, where the line crosses the y-axis.

Linear relationships are important because they provide a straightforward way to predict one variable based on another. By transforming non-linear equations, like exponential functions, into linear forms, patterns become easier to interpret and visualize. This is especially useful in data analysis and scientific research.

Log-Linear Plot

A log-linear plot is a type of graph used to visualize equations that have been linearized through logarithmic transformations. To create a log-linear plot, one axis of the graph is on a logarithmic scale, while the other is on a linear scale. When plotting an exponential function, like \( v = 2e^{-1.2x} \), as a linear equation after taking the natural logarithm (\( \ln(v) \)), the plot becomes a straight line. This is because the logarithmic transformation transforms the exponential decay or growth into a slope. On a log-linear plot:

• The slope of the line reflects the rate of change or decay.
• The intercept provides specific values that can help understand the original equation.

Using log-linear plots helps in detecting trends over wide-ranging data sets, exploring patterns in data, and simplifying complex relationships for analysis.

Natural Logarithm

The natural logarithm is a special type of logarithm with a base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. The natural logarithm is often written as \( \ln(x) \), and it is extensively used in science and mathematics to handle exponential growth or decay problems.The properties of natural logarithms include:

• \( \ln(1) = 0 \), because \( e^0 = 1 \).
• \( \ln(e) = 1 \), as \( e^1 = e \).
• \( \ln(a \,b) = \ln(a) + \ln(b) \), showing the logarithmic identity for multiplication.
• \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \) for division.

These properties allow us to simplify complex equations, such as turning an exponential equation into a linear one by taking logarithms, making it easier for analysis and understanding.

Exponential Function

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. The general form is \( a^x \), where \( a \) is the base and \( x \) is the exponent. These functions are known for their rapid growth or decay and describe numerous natural phenomena.For instance, the function \( v = 2e^{-1.2x} \) describes exponential decay, because the exponent \(-1.2x\) is negative. Key characteristics of exponential functions include:

• They have a rapid rate of change and can describe processes such as population growth, radioactive decay, and more.
• The graph of an exponential function is a smooth curve that increases or decreases steadily.
• By transforming an exponential function using logarithms, you convert it into a linear form for easier interpretation.

Understanding these functions is crucial in modeling real-world scenarios and making predictions based on mathematical equations.