Problem 50 - 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. $$ y=7 e^{3 x} $$

Step-by-Step Solution

Answer

Use \( \ln(y) = 3x + \ln(7) \) to plot a linear graph of \( \ln(y) \) vs. \( x \).

Step 1: Identify the Relationship

The given equation is \( y = 7 e^{3x} \) where \( y \) is expressed in terms of \( x \). We want to transform this into a linear relationship.

Step 2: Apply the Logarithmic Transformation

To linearize the equation, apply the natural logarithm to both sides of the equation to obtain \( \ln(y) = \ln(7 e^{3x}) \).

Step 3: Use Logarithm Properties

Use the property \( \ln(a \times b) = \ln(a) + \ln(b) \) to rewrite \( \ln(7 e^{3x}) \) as \( \ln(7) + \ln(e^{3x}) \).

Step 4: Simplify the Equation

Since \( \ln(e^{3x}) = 3x \), substitute this into the equation to get \( \ln(y) = \ln(7) + 3x \).

Step 5: Identify the Linear Form

The equation \( \ln(y) = \ln(7) + 3x \) is now in the form \( Y = mx + C \), where \( Y = \ln(y) \), \( m = 3 \), and \( C = \ln(7) \). This is a linear relationship.

Step 6: Graph the Linear Relationship

To graph this linear relationship, plot \( Y \) (which is \( \ln(y) \)) on the vertical axis against \( x \) on the horizontal axis. This will be a straight line with slope 3 and y-intercept \( \ln(7) \) on a log-linear plot.

Key Concepts

Log-Linear Plots

In mathematics, a log-linear plot is an immensely useful tool when dealing with exponential relationships. It transforms what would typically be a curve into a straight line, simplifying analysis. By using logarithms, we make nonlinear relationships linear, making it easier to interpret data patterns and predict trends. When graphing exponential functions like our given equation, a log-linear plot involves plotting the logarithm of the function's output (ln(y)) against the input variable (x).
In the exercise, we transformed the equation \( y = 7e^{3x} \) to \( \ln(y) = \ln(7) + 3x \). This new equation is linear, making it straightforward to graph. This graph showcases a clear linear relationship on a log-linear plot with a slope of 3 and a y-intercept of \( \ln(7) \).
Such plots are particularly helpful in fields such as biology, economics, and physics, where exponential growth and decay patterns frequently occur.

• Easier comparison of rates of change
• Visibility of consistent growth patterns
• Simplification of exponential relationships

Exponential Functions

Exponential functions are fundamental in mathematics and are observed in various real-world scenarios. They have the form \( y = a \cdot e^{bx} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.718. The defining feature of exponential functions is their rapid growth or decay, depending on the sign of \( b \).
In the context of our exercise, \( y = 7 e^{3x} \) represents an exponential function where the base \( e \) is raised to the power of \( 3x \), indicating growth because the exponent is positive.\( a = 7 \) indicates the initial value, providing the function's starting point. Exponential functions are vastly used in modeling population dynamics, radioactive decay, and financial growth predictions.
Key characteristics of exponential functions include:

• Continuous and smooth graph
• Horizontal asymptote (usually the x-axis)
• Rapid increase or decrease
• Constant percentage rate of growth or decay

Linearization in Calculus

Linearization is a useful calculative technique to approximate complex functions with simpler linear equations. It allows us to find a linear function that closely fits the curve of a nonlinear function at a specific point or interval. This is particularly handy when evaluating or predicting behavior in a small neighborhood around a point.
By applying the logarithmic transformation, as shown in our exercise, we took the nonlinear exponential function \( y = 7e^{3x} \) and converted it into the linear form \( \ln(y) = \ln(7) + 3x \). This process is directly related to linearization, simplifying the understanding and visualization of the function.
Important points about linearization include:

• Approximates nonlinear functions near a specified point
• Simplifies calculations and predictions
• Widely used in engineering and the sciences for analysis
• Improves insight into local behavior of functions