Problem 66 - 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 o g ~ p l o t . ~ $$ y=9 x^{-3} $$

Step-by-Step Solution

Answer

The linear relationship is \( \ln(y) = -3\ln(x) + \ln(9) \), a line with slope -3 on the log-log plot.

Step 1: Understand the Problem

We are given an equation: \( y = 9x^{-3} \). Our task is to apply a logarithmic transformation to both sides of this equation to find a linear relationship suitable for plotting on a log-log graph.

Step 2: Apply Logarithmic Transformation

Take the natural logarithm of both sides of the equation. This gives us: \( \ln(y) = \ln(9x^{-3}) \). Using logarithm properties, we can separate this into: \( \ln(y) = \ln(9) + \ln(x^{-3}) \).

Step 3: Simplify the Logarithm

Utilize the power rule for logarithms: \( \ln(x^{-3}) = -3\ln(x) \). Substitute this back into our equation: \( \ln(y) = \ln(9) - 3\ln(x) \).

Step 4: Identify the Linear Relationship

Rewrite the equation in the form of \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept: \( \ln(y) = -3\ln(x) + \ln(9) \). This represents a linear equation with slope \( -3 \) and y-intercept \( \ln(9) \).

Step 5: Prepare for Log-Log Plot

To create a log-log plot: the x-axis will represent \( \ln(x) \) and the y-axis will represent \( \ln(y) \). From the linear equation \( \ln(y) = -3\ln(x) + \ln(9) \), plot the line where slope \( m = -3 \) and y-intercept \( \ln(9) \).

Key Concepts

Understanding Log-Log Plot

A log-log plot is a special type of graph used to understand relationships between variables that span several orders of magnitude. When both the x-axis and y-axis of a graph are logarithmic scales, it's called a log-log plot. This kind of plot is particularly useful when dealing with power laws or exponential functions.

In a log-log plot, you don't directly plot the actual values of x and y. Instead, you plot their logarithmic values. For instance, instead of x, you use \( \ln(x) \) and for y, you use \( \ln(y) \).

This is significant when the relationships between the variables are multiplicative. Log-log plots can turn a complex curve into a simple straight line, making it easier to analyze and interpret the data:

• Identify power-law relationships, where the form is y = kx^n.
• Establish the proportionality and scaling between variables.
• Simplify the view of large ranges of data.

So in our example, plotting \( \ln(y) = -3\ln(x) + \ln(9) \) on a log-log plot creates a straight line. This simplifies analysis, as we can easily read the slope and intercept from the graph.

Deciphering Linear Relationships

Linear relationships are represented by the equation of a line, \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. Identifying a linear relationship is crucial in simplifying complex equations.

In the context of logarithmic transformations, once transformed, the equation often reveals a linear form. For instance, our original equation \( y = 9x^{-3} \) transforms into \( \ln(y) = -3\ln(x) + \ln(9) \). This equation is linear with respect to the log values, where:

• The slope \( m \) is -3. This tells us that as \( x \) increases, \( y \) decreases at a proportional rate.
• The y-intercept \( c \) is \( \ln(9) \), giving us the starting point on our log-log graph.

Recognizing these linear patterns can transform how we understand data across various fields, such as physics, chemistry, and economics, where relationships between quantities often adhere to power laws or exponential growth patterns.

The Magic of Exponential Functions

Exponential functions have the general form of \( y = a \cdot b^x \), where \( a \) is a constant and \( b \) is the base of the exponential. These functions are powerful because they describe rapid changes and are prevalent in natural phenomena.

In the context of our exercise, we began with an inverse power function \( y = 9x^{-3} \). Although it doesn't fit the traditional exponential form directly, the negative exponent and the subsequent transformation through logarithms help in comprehending its exponential nature.

Understanding exponential functions is essential for:

• Modeling growth and decay processes, like population growth or radioactive decay.
• Analyzing economic trends, such as compound interest.
• Interpreting natural phenomena, like light attenuation and sound intensity.

By turning the exponential relationship into a linear one using logarithms, we simplify analysis, making it more visually intuitive and mathematically straightforward. This also allows us to predict future behaviours of systems modelled by these functions.