Problem 47 - 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 . ~ $$ y=3 \times 10^{-2 x} $$

Step-by-Step Solution

Answer

The linear relationship is \( \ln(y) = \ln(3) - 2\ln(10)x \), representing a line on a log-linear plot.

Step 1: Understand the Given Equation

The given function is \( y = 3 \times 10^{-2x} \). Here, \( y \) is expressed in terms of the exponential base power function involving \( x \). Our goal is to transform this into a linear relationship using logarithms.

Step 2: Apply Logarithmic Transformation

To linearize the equation, apply the natural logarithm on both sides of the equation: \( \ln(y) = \ln(3 \times 10^{-2x}) \). Using logarithmic properties, this becomes: \( \ln(y) = \ln(3) + \ln(10^{-2x}) \). Further, the property \( \ln(a^b) = b \ln(a) \) allows us to write: \( \ln(10^{-2x}) = -2x\ln(10) \). Thus, the equation simplifies to \( \ln(y) = \ln(3) - 2x\ln(10) \).

Step 3: Identify the Linear Equation

Re-writing the transformed equation, we get \( \ln(y) = \ln(3) - 2\ln(10)x \). This is of the form \( y = mx + b \) where \( m = -2\ln(10) \) and \( b = \ln(3) \). This is a linear equation in the form of \( \ln(y) \) versus \( x \).

Step 4: Plot the Linear Relationship

The linear relation derived, \( \ln(y) = \ln(3) - 2\ln(10)x \), can be plotted on a log-linear plot with \( \ln(y) \) on the y-axis and \( x \) on the x-axis. The slope of the line is \(-2\ln(10)\) reflecting the inverse relationship and \( \ln(3) \) as the intercept on the y-axis.

Key Concepts

Linear Relationship

A linear relationship describes a straight-line connection between two variables. It signifies that when one variable, say "x," changes, the other variable "y" changes at a constant rate. In mathematical terms, this can be represented with the equation of a line:

• y = mx + b

Here, "m" represents the slope of the line, which indicates how steep the line is, and "b" is the y-intercept, the point where the line crosses the y-axis.
In our problem, we started with an exponential function and converted it into a linear form using logarithmic transformations. This resulted in a format similar to a linear equation, with constants derived from the logarithm of the original coefficients. This illustrates how exponential relationships can be reinterpreted as linear ones through logarithmic transformation, making complex relationships easier to analyze and understand.

Log-Linear Plot

A log-linear plot is a type of graph used to display data that follows an exponential trend. It is especially useful for visualizing relationships where one variable changes exponentially with respect to another.
In this kind of plot, the x-axis is typically linear, while the y-axis is logarithmic. By plotting the logarithm of a variable on the y-axis, an exponential relationship can appear as a straight line. This method allows us to observe trends and make interpretations easier.
The transformed equation from our problem,

• \( \ln(y) = \ln(3) - 2\ln(10) x \)

can be plotted using a log-linear plot. Here, one would plot \( \ln(y) \) against \( x \), resulting in a straight line whose slope is

• \(-2\ln(10)\)

This visual representation helps in identifying patterns that would be less obvious in non-linear plotting. Understanding how to read log-linear plots is beneficial for interpreting different types of data and revealing underlying patterns.

Exponential Functions

Exponential functions are mathematical expressions involving a constant base raised to a variable exponent. They are characterized by their rapid rate of change, which makes them valuable in modeling growth and decay processes.
A typical exponential function can be expressed in the form:

• \( y = a \times b^{x} \)

where "a" is the initial amount, "b" is the base representing a growth factor or decay factor, and "x" is the exponent.
The function from the problem,

• \( y = 3 \times 10^{-2x} \)

illustrates a scenario where y decreases exponentially as x increases due to the negative exponent. Transforming exponential functions using logarithms, as shown, can effectively convert them into linear relationships. This is particularly useful for understanding and forecasting patterns in fields like economics, biology, and physics.