Problem 70 - Calculus for Biology and Medicine

Question

Use a logarithmic transformation to find a linear relationship between the given quantities and determine whether a log-log or log-linear plot should be used to graph the resulting linear relationship. $$ I(u)=4.8 u^{-0.89} $$

Step-by-Step Solution

Answer

Use a log-log plot, as both \( I(u) \) and \( u \) were logarithmically transformed for linearity.

Step 1: Understanding the Problem

First, we identify that the exercise requires finding a linear relationship using a logarithmic transformation of the given function \( I(u)=4.8 u^{-0.89} \). The goal is to transform the equation so it produces a straight line when plotted.

Step 2: Apply Logarithmic Transformation

Take the logarithm of both sides of the equation. \[\log(I(u)) = \log(4.8 \, u^{-0.89})\]

Step 3: Use Logarithmic Properties

Use the properties of logarithms, particularly \( \log(ab) = \log(a) + \log(b) \) and \( \log(a^b) = b \log(a) \): \[\log(I(u)) = \log(4.8) + \log(u^{-0.89}) = \log(4.8) - 0.89 \log(u)\]

Step 4: Identify the Linear Equation

The equation \( \log(I(u)) = \log(4.8) - 0.89 \log(u) \) is in the form \( Y = B + MX \), where:- \( Y = \log(I(u)) \)- \( X = \log(u) \)- \( M = -0.89 \) is the slope - \( B = \log(4.8) \) is the y-interceptThis identifies a linear relationship when plotting \( \log(I(u)) \) against \( \log(u) \).

Step 5: Determine Plot Type

Since the transformation involves taking the logarithm of both variables \( I(u) \) and \( u \), a log-log plot should be used to graph the linear relationship.

Key Concepts

Exploring the Log-Log Plot

To understand the magic of turning certain equations into straight lines, we often use a log-log plot. This involves plotting the logarithms of both the dependent and independent variables on the axes. Imagine taking the given equation \( I(u) = 4.8 u^{-0.89} \) and transforming it into a more manageable form. When both the y-values (\(I(u)\)) and x-values (\(u\)) are transformed using a logarithm, you simplify the complexity of the equation.

• This transformation helps showcase a linear relationship between the variables, often hidden in their original form.
• In our example, by plotting \( \log(I(u)) \) against \( \log(u) \), the data will align to form a straight line.

Thus, a log-log plot is particularly powerful when dealing with data that follows a power law, as seen in the equation \( I(u) = 4.8 u^{-0.89} \). This is why scientists and engineers often rely on log-log plots to interpret data and draw conclusions.

Understanding the Linear Relationship

A linear relationship means that as one variable changes, the other changes in a directly proportional way, and this can often be expressed with the equation \( Y = MX + B \). In our task, the logarithmic transformation helped us identify such a relationship in the equation \( \log(I(u)) = \log(4.8) - 0.89 \log(u) \). Here's how:

• With \( Y = \log(I(u)) \) and \( X = \log(u) \), you've set up the problem to be analyzed as a straight line.
• The slope, denoted by \( M \) here, is \(-0.89\), indicating the rate of change.
• The y-intercept \( B \) is \( \log(4.8) \), which is where the line would cross the y-axis when the line is extended to the x-axis (\(X = 0\)).

When we express relationships that way, it simplifies the interpretation and prediction of data trends, making analysis easier and insights clearer.

Decoding Logarithmic Properties

Logarithmic properties serve as essential tools for simplifying equations and uncovering hidden patterns. Let's break down the crucial properties used in our transformation:

• The Product Rule: \( \log(ab) = \log(a) + \log(b) \) breaks down multiplication inside a log into a more manageable sum of logs.
• The Power Rule: \( \log(a^b) = b \log(a) \) transforms an exponent into a multiplied coefficient, simplifying exponentials within logarithms.
• Using these properties with our equation \( \log(4.8u^{-0.89}) \) results in an expression of \( \log(4.8) - 0.89 \log(u) \).

Understanding these properties enables one to make calculations more practical and facilitate the transition from non-linear to linear relationships. This is foundational for converting complex, real-world problems into simpler, analyzable forms.