Problem 65 - 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 . ~ $$ y=4 x^{-3} $$

Step-by-Step Solution

Answer

The relationship is linear on a log-log plot, with slope -3 and y-intercept \(\log(4)\).

Step 1: Understand the Problem

We need to transform the given equation \( y = 4x^{-3} \) into a linear form using logarithms, and then represent this on a log-log graph. This involves expressing both sides of the equation in terms of logarithms.

Step 2: Apply Logarithmic Transformation

Firstly, let's apply the logarithm to both sides of the equation. Using the property \( \log(a^b) = b\log(a) \), we can rewrite the equation as follows: \[\log(y) = \log(4x^{-3})\] This can be expanded further using logarithm properties to: \[\log(y) = \log(4) + \log(x^{-3})\] Which becomes: \[\log(y) = \log(4) - 3\log(x)\]

Step 3: Rearrange into Linear Form

The expression \( \log(y) = \log(4) - 3\log(x) \) can be rearranged into the standard linear form \( y = mx + b \), where:- \( y \) is \( \log(y) \)- \( x \) is \( \log(x) \)- The slope \( m = -3 \)- The y-intercept \( b = \log(4) \).Thus, the linear equation is:\[\log(y) = -3\log(x) + \log(4)\]

Step 4: Plot the Linear Relationship

To create the log-log plot, plot \( \log(y) \) against \( \log(x) \). The plot will be a straight line due to the linear relationship, where the slope is \(-3\) and the y-intercept is \( \log(4) \).

Key Concepts

Linear Relationship

A linear relationship describes a straight-line connection between two variables. In other words, changes in one variable lead to proportional changes in the other. The general equation for a linear relationship is in the form of

• \[ y = mx + b \]
• where m is the slope and b is the y-intercept.

By transforming a non-linear equation like

• \[ y = 4x^{-3} \]
• into a linear one through logarithmic operations, we identify the values of the slope \( m \) and intercept \( b \) for a clearer analytical view.

This enables us to understand how the variables

• logarithm of y and
• logarithm of x relate linearly,

turning the complex power law into a more manageable linear framework.This transformation offers a simplified approach to grasping the concept and applying it to real-world scenarios, where relationships are often not initially apparent.

Log-Log Plot

A log-log plot is an invaluable tool when dealing with power-law relationships. This type of graph plots the logarithm of one variable against the logarithm of another. By doing this, it linearizes exponential and power-law data.

• For example, the transformation of
• \[ y = 4x^{-3} \]

yields a linear equation:

• \[ \log(y) = -3\log(x) + \log(4) \]

In the resulting log-log plot:

• - The x-axis represents \( \log(x) \),
• - The y-axis represents \( \log(y) \),
• - The slope of the line is -3,
• - The y-intercept is \( \log(4) \).

This method allows for simplified visual analysis, making it easy to determine relationships between variables that appear complex at first glance. It turns non-linear curves into linear lines, easing the interpretation and gaining insights from data.

Calculus

Calculus is a branch of mathematics that studies continuous change. While calculus provides tools like derivatives and integrals, in the context of logarithmic transformation, its core concepts help us understand variable relationships and their rates of change.
When dealing with equations that undergo transformation, logarithms play a significant role in simplifying these relationships.

• The idea of transforming
• \[ y = 4x^{-3} \]

into a linear relation exemplifies how calculus concepts intertwine with logarithmic functions.Through the lens of calculus, logarithmic transformations help unveil hidden variable relationships and enable us to see patterns in complex data by expressing it linearly.
This is because calculus offers a broad framework to explore the behavior of functions, different rates of change, and the relationships among varying quantities.