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

A log-log plot should be used for this transformation.

Step 1: Identify the Equation

We are given the equation for intensity, \[ I(u) = 4.8 u^{-0.89}. \] Our task is to find whether a log-log or log-linear relationship will linearize this equation.

Step 1: Take the Logarithm of Both Sides

Apply the natural logarithm to both sides of the equation \( I(u) = 4.8 u^{-0.89} \) to begin transforming it.\[ \ln(I) = \ln(4.8 u^{-0.89}). \]

Key Concepts

Linear Relationship

A linear relationship between two quantities exists when a change in one results in a proportional change in the other. In simple terms, if you plot the two variables on a graph, the points fall on a straight line. This is a simple and fundamental concept in mathematics and statistics. A linear relationship is often described by the equation of a line, which is generally written as

• \( y = mx + c \)

where \( y \) and \( x \) are the variables, \( m \) is the slope (or gradient) of the line, and \( c \) is the y-intercept.
In the context of transformations, making an equation linear enables us to use statistical techniques efficiently.
For example, with logarithmic transformations, if a quantity varies exponentially or is a power of another, applying a logarithm can convert this into a linear relationship. This is particularly useful in data analysis and modeling, allowing predictions and insights that would be obscured by more complex relationships.

• The goal of transformations is to simplify analysis.
• Linear relationships make it easier to understand the relationship between variables.

Log-Log Plot

A log-log plot is used when both the x-axis and the y-axis use logarithmic scales. This kind of plot is particularly useful when examining relationships between two quantities that expand over several orders of magnitude.
To use a log-log plot effectively, both variables in the equation need to be transformed using the logarithm function.
For example, if we have an equation like

• \( I(u) = 4.8 u^{-0.89} \)

we can take the log of both sides:

• Step 1: \( \ln(I) = \ln(4.8) + \ln(u^{-0.89}) \)
• Step 2: Simplify to \( \ln(I) = \ln(4.8) - 0.89 \ln(u) \)

This reveals a linear form:

• \( y = mx + c \) with \( y = \ln(I) \), \( m = -0.89 \), \( x = \ln(u) \), and \( c = \ln(4.8) \)

Log-log plots transform exponential growth into linear growth, which can be vital for analyzing scaling laws or power laws in data science, physics, and other fields.

Log-Linear Plot

A log-linear plot features one axis (usually the y-axis) on a logarithmic scale, while the other axis remains on a linear scale. This type of plot is typically used when one quantity changes exponentially in response to the other.
To create a log-linear plot for an equation:

• We keep the x-variable linear.
• We transform the y-variable via logarithms.

Consider a relationship like:

• \( I(u) = 4.8 u^{-0.89} \)

Taking the logarithm of the dependent variable:

• \( \ln(I) = \ln(4.8) - 0.89 \ln(u) \)

However, if the equation had been in a form like

• \( I(u) = 4.8 e^{-0.89u} \)

this would result in a linear relationship directly, with \( \ln(I) \) as the dependent variable plotted against the linear \( u \) on the x-axis.
Log-linear plots are powerful in identifying exponential trends and are widely used in fields involving growth rates, such as biology, finance, or economics.