Problem 60 - Calculus for Biology and Medicine

Question

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-log plot. $$ y=3 x^{2} $$

Step-by-Step Solution

Answer

The linear relationship is \( \log(y) = 2\log(x) + \log(3) \). It will appear as a straight line with slope 2 on a log-log plot.

Step 1: Identify the Given Equation

The given equation is \( y = 3x^2 \). We need to transform this into a linear equation that can be plotted on a log-log graph.

Step 2: Apply Logarithmic Transformation

Take the logarithm of both sides of the equation. This gives: \( \log(y) = \log(3x^2) \). Using the properties of logarithms, we can separate it as \( \log(y) = \log(3) + \log(x^2) \).

Step 3: Simplify the Logarithmic Expression

Apply the power rule of logarithms: \( \log(x^2) = 2 \log(x) \). Substituting this in, the equation becomes: \( \log(y) = \log(3) + 2 \log(x) \).

Step 4: Express in Linear Form

Rewrite the transformed equation to resemble the linear form \( y = mx + b \). We have \( \log(y) = 2\log(x) + \log(3) \). Here, \( \log(y) \) is analogous to \( y \), \( 2 \) as the slope \( m \), \( \log(x) \) as \( x \), and \( \log(3) \) as the y-intercept \( b \).

Step 5: Graph the Linear Relationship

On a log-log plot, the x-axis and y-axis are logarithmic. Plot \( \log(y) \) against \( \log(x) \). The graph will be a straight line with a slope of 2 and a y-intercept of \( \log(3) \).

Key Concepts

Understanding Linear Equations

Linear equations are essential in mathematics. They describe relationships involving a constant rate of change between two variables. The general form of a linear equation is \( y = mx + b \), where:

• \( y \) is the dependent variable.
• \( x \) is the independent variable.
• \( m \) represents the slope, or the rate of change.
• \( b \) is the y-intercept, the value of \( y \) when \( x = 0 \).

Linear equations are the foundation for more complex mathematical concepts. They provide a simple framework that helps in data modeling and analysis. By understanding linear equations, you can analyze patterns and predict trends. This is why transforming other types of equations, like quadratics, into a linear form is very useful. It allows you to use linear models to interpret the data and relationships.

Exploring Logarithm Properties

Logarithms simplify complex equations by transforming multiplicative relationships into additive ones. They come with a set of properties that make calculations easier:

• Product Rule: \( \log(ab) = \log(a) + \log(b) \). This rule allows you to break down the product of two numbers into a sum of their logs.
• Quotient Rule: \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \). This simplifies the division of two numbers into a difference of their logs.
• Power Rule: \( \log(a^b) = b \log(a) \). This is particularly helpful in our transformation, as it allows us to easily handle powers.

Using these properties, complex equations such as exponential or power functions can be transformed into more manageable linear forms. For example, in the original equation \( y = 3x^2 \), the power rule simplifies \( \log(x^2) \) to \( 2\log(x) \), making it easier to interpret and graph.

Understanding Log-Log Plots

A log-log plot is a graph that uses logarithmic scales for both the x and y axes. This kind of plot is useful for understanding relationships that follow a power law, where one variable is proportional to a power of another variable. Let's explore further:

• Logarithmic Scale: On a log-log plot, each unit increase on the axis represents a ten-fold increase. This is essential for accurately representing wide-ranging data.
• Linear Relationship: A power-law relationship, after transformation, appears as a straight line on a log-log plot, where the slope corresponds to the exponent in the original equation.
• Visualization: Transforming data into a log-log plot helps reveal trends and relationships that aren’t immediately apparent on standard linear plots.

For the equation \( y = 3x^2 \), by plotting \( \log(y) \) versus \( \log(x) \), we get a straight line with a slope of 2. This confirms the relationship and makes it easy to interpret large or complex datasets.