Problem 63 - 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 . ~ $$ y=2 x^{-2} $$

Step-by-Step Solution

Answer

Plot \( \ln(y) = \ln(2) - 2\ln(x) \) on a log-log plot for a linear relationship.

Step 1: Understanding the Equation

We start with the given equation \( y = 2x^{-2} \). Our goal is to transform this into a linear relationship suitable for a log-log plot.

Step 2: Apply Logarithm to Both Sides

Take the natural logarithm of both sides: \( \ln(y) = \ln(2x^{-2}) \). Use properties of logarithms to expand this: \( \ln(y) = \ln(2) + \ln(x^{-2}) \).

Step 3: Simplify the Logarithm

Utilize the power rule for logarithms where \( \ln(x^{-a}) = -a \ln(x) \). Thus, \( \ln(x^{-2}) = -2\ln(x) \). Our equation becomes \( \ln(y) = \ln(2) - 2\ln(x) \).

Step 4: Identify Linear Relationship

The equation \( \ln(y) = \ln(2) - 2\ln(x) \) is now in the form \( \ln(y) = c - m\ln(x) \) where \( c = \ln(2) \) represents the y-intercept and \( m = 2 \) is the slope. This is a linear relationship in a log-log space.

Step 5: Plot the Linear Relationship on a Log-Log Plot

On a log-log plot, the axes are \( \ln(x) \) and \( \ln(y) \). Plot \( \ln(y) \) against \( \ln(x) \), and it will yield a straight line with a slope of \(-2\) and y-intercept \( \ln(2) \).

Key Concepts

Log-log Plot

A log-log plot is a type of graph where both axes are on a logarithmic scale. This means that each tick mark on the axes represents the logarithm of a data point rather than the data point itself. Such plots are incredibly useful when you're dealing with data that follows a power law or when transforming nonlinear relationships into linear ones.

By transforming your data logarithmically, you can plot equations like power functions \( y = ax^b \) as straight lines. This makes it much easier to identify relationships and interpret the data.

For instance, if you have an equation like \( y = 2x^{-2} \), by taking the logarithm of both sides, you can transform this into a linear equation, which is\( \ln(y) = \ln(2) - 2 \ln(x) \). When you plot \( \ln(y) \) versus \( \ln(x) \) on a log-log plot, you will see a straight line with a slope indicating the power of the relationship, \(-2\).

In summary, log-log plots simplify the complexity of power relationships to make analysis more straightforward.

Linear Relationship

A linear relationship is a type of relationship in which the change in the dependent variable is proportional to the change in the independent variable. In other words, when you plot this type of relationship on a graph, you get a straight line. This is characterized by the equation \( y = mx + c \), where \( m \) is the slope of the line, and \( c \) is the y-intercept.

In the context of logarithmic transformations, linearizing a non-linear relationship involves using the properties of logarithms to turn the input (usually a power function) into a straight line that can be easily interpreted. For example, starting with a power equation like \( y = 2x^{-2} \), taking the natural log of both sides allows us to linearize it as \( \ln(y) = \ln(2) - 2\ln(x) \). The transformed equation is linear because it can be written in the form \( \ln(y) = c - m\ln(x) \), where it resembles the standard formula of \( y = mx + c \).

This linear format is easier to analyze, as the slope \( m \) directly tells you about how \( x \) affects \( y \), while the intercept \( c \) provides the initial value when \( x = 1 \). Linearization through logs hence enables us to employ statistical tools and techniques typically reserved for linear data.

Power Rule for Logarithms

The power rule for logarithms is a crucial concept when transforming equations for analysis. This rule states that \( \ln(a^b) = b \ln(a) \) as long as \( a > 0 \). This means you can bring the exponent down in front of the logarithm as a multiplier. It's particularly useful for simplifying the logs of expressions where a variable is raised to a power.

In practice, when you have an equation like \( y = 2x^{-2} \) and you take the log of both sides, you apply the power rule to simplify: \( \ln(y) = \ln(2) + \ln(x^{-2}) \). Using the power rule, this becomes \( \ln(x^{-2}) = -2\ln(x) \). The equation is simplified to \( \ln(y) = \ln(2) - 2\ln(x) \), turning your initial expression into a format suitable for linear regression on a log-log plot.

This conversion is key to working with power functions, making complex equations manageable and readily analyzable by stripping them to fundamental linear relationships. Understanding and applying the power rule allows for robust analytical simplifications.