Problem 62 - 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

The logarithmic transformation yields a linear equation \( \log(y) = -2\log(x) + \log(2) \) with a slope of \(-2\).

Step 1: Identify the given equation

The given equation is \( y = 2x^{-2} \). This implies that \( y \) is inversely proportional to the square of \( x \). Our goal is to convert this equation into a linear form using logarithms.

Step 2: Apply the logarithmic transformation

To linearize the relationship, take the logarithm of both sides of the equation. This gives: \( \log(y) = \log(2x^{-2}) \). Using logarithmic properties, this can be expanded to \( \log(y) = \log(2) + \log(x^{-2}) \).

Step 3: Simplify using logarithm properties

Use the property \( \log(x^a) = a\log(x) \) to simplify \( \log(x^{-2}) \) to \(-2\log(x) \). Thus, the equation becomes \( \log(y) = \log(2) - 2\log(x) \).

Step 4: Express the equation in a linear form

Rearrange the equation \( \log(y) = \log(2) - 2\log(x) \) to show its linear form: \( \log(y) = -2\log(x) + \log(2) \). This is in the form \( y = mx + c \), where \( m = -2 \) and \( c = \log(2) \).

Step 5: Interpret the linear function

The linear equation \( \log(y) = -2\log(x) + \log(2) \) suggests that on a log-log plot, the relationship between \( x \) and \( y \) will be a straight line with slope \(-2\) and a y-intercept of \( \log(2) \).

Step 6: Graph the linear relationship on a log-log plot

To visualize this, plot \( \log(y) \) on the y-axis and \( \log(x) \) on the x-axis. The line should have a slope of \(-2\), indicating the relationship \( y \approx x^{-2} \), and intersect the y-axis at \( \log(2) \).

Key Concepts

linear relationship

In mathematics, a linear relationship refers to a relationship between two variables where one variable is a constant multiple of the other, resulting in a straight line when graphed on a coordinate plane. This relationship is often expressed as the equation of a line, typically written in the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

Understanding linear relationships is fundamental because it helps us see patterns and predict values. For example, if you know the slope and intercept, you can efficiently determine \(y\) for any given \(x\).

• The slope \(m\) indicates the rate of change between the variables. A positive slope means that as \(x\) increases, \(y\) also increases. A negative slope, as in our example of \(y = -2\log(x) + \log(2)\), indicates that \(y\) decreases as \(x\) increases.
• The intercept \(c\) tells us where the line crosses the y-axis, i.e., the value of \(y\) when \(x = 0\).

Linear relationships are a crucial concept not just in mathematics but also in science, economics, and various fields that rely on modeling and data analysis.

log-log plot

A log-log plot is a graphical representation where both axes are on a logarithmic scale. This allows us to transform nonlinear relationships into linear ones, making it easier to identify and interpret patterns and trends between the variables. In the case of the exercise, we converted the equation \(y = 2x^{-2}\) into a linear form with logarithms.

In log-log plots, the data is expressed in terms of logarithmic scales: both the x-axis and y-axis display logarithmic values of the original dataset. This transformation can demystify relationships that appear nonlinear in cartesian coordinates.

• It is particularly useful for relationships that involve powers, such as polynomial and power-law relationships, where the transformed data reveals a linear pattern.
• By plotting the transformation \(\log(y) = -2\log(x) + \log(2)\), we obtain a straight line with a slope of \(-2\) and a y-intercept of \(\log(2)\).

This allows us to understand the proportional relationships more clearly and compare the magnitudes of different datasets efficiently. Log-log plotting is a powerful tool for visualizing changes that occur exponentially over multiple scales.

inverse proportionality

An inverse proportionality describes a situation where one variable increases while the other decreases at a constant rate. For the equation \(y = 2x^{-2}\), \(y\) is inversely proportional to the square of \(x\). This means as \(x\) grows, \(y\) decreases, and as \(x\) diminishes, \(y\) increases.

In terms of relationships, if \(y\) and \(x\) are inversely proportional, the product \(xy\) remains constant. In our exercise, when \(x\) is squared and inversely related, this changes the directness of the relationship. This direct relationship can appear more complex and, in many real-world applications, we simplify by expressing them through transformations like logarithms to better grasp this inverse nature.

• The concept of inverse proportionality is crucial in physics and chemistry, where it often appears in laws describing natural phenomena, like Newton's law of universal gravitation or Boyle's law.
• Recognizing inverse relationships helps in constructing and deconstructing mathematical models to predict outcomes accurately.

By employing logarithmic transformation, these relationships can be recast to reveal linear patterns that simplify our understanding and interpretation of inverse proportional connections.