Problem 71 - 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. $$ R(t)=3.6 t^{1.2} $$

Step-by-Step Solution

Answer

Use a log-log plot for \( R(t) = 3.6 t^{1.2} \); the linear form is \( \ln(R(t)) = 1.2\ln(t) + \ln(3.6) \).

Step 1: Understand the Problem

We need to convert the function \( R(t) = 3.6 t^{1.2} \) into a linear form using logarithms. A linear function can be expressed as a straight line, which is not the current form of \( R(t) \).

Step 2: Apply Logarithmic Transformation

To linearize the given function, take the logarithm of both sides. Applying the natural logarithm, we have: \[ \ln(R(t)) = \ln(3.6 t^{1.2}) \].

Step 3: Use Logarithmic Properties

Use the properties of logarithms where \( \ln(ab) = \ln(a) + \ln(b) \) and \( \ln(b^c) = c\ln(b) \) to simplify the equation: \[ \ln(R(t)) = \ln(3.6) + \ln(t^{1.2}) = \ln(3.6) + 1.2\ln(t) \].

Step 4: Rewrite in Linear Form

Rearrange the equation to make it appear in the form of \( y = mx + c \): \[ y = 1.2 x + \ln(3.6) \], where \( y = \ln(R(t)) \) and \( x = \ln(t) \). This is now in the equation of a straight line with slope 1.2 and y-intercept \( \ln(3.6) \).

Step 5: Determine Plot Type

Since both the response variable \( \ln(R(t)) \) and the explanatory variable \( \ln(t) \) are in logarithmic form, we should use a log-log plot to graph this linear relationship.

Key Concepts

Linear Relationship

A linear relationship describes a straight-line connection between two variables. In mathematical terms, it’s expressed as \( y = mx + c \), where \( m \) is the slope, \( x \) is the independent variable, \( y \) is the dependent variable, and \( c \) is the y-intercept. This formula means as the independent variable changes, the dependent variable changes at a constant rate.

In this exercise, our task was to transform the function \( R(t) = 3.6 t^{1.2} \) to showcase a linear form through a logarithmic transformation. By taking the natural logarithm of both sides of the equation, we made it resemble the linear equation format. The transformation resulted in \( \ln(R(t)) = 1.2 \ln(t) + \ln(3.6) \). Here, \( y = 1.2x + c \) is clearly visible, with \( y \) being \( \ln(R(t)) \) and \( x \) being \( \ln(t) \).

Understanding these relationships is crucial for analyzing how one variable can predict another, which is especially useful in statistical modeling and forecasting.

Log-Log Plot

A log-log plot is a type of graphical representation where the axes are scaled logarithmically. This scaling is particularly useful when dealing with power relations, like \( R(t) = 3.6 t^{1.2} \), as it can linearize the power function into a straight line.

In this exercise, after applying logarithmic transformation, we ended up with \( \ln(R(t)) = 1.2 \ln(t) + \ln(3.6) \). By plotting \( \ln(t) \) on the x-axis and \( \ln(R(t)) \) on the y-axis, we get a straight line whose slope is 1.2. This confirms a power-law relationship between \( R(t) \) and \( t \).

Log-log plots are particularly helpful because they can make exponential growth appear linear, depicting proportional changes between variables more explicitly. This type of analysis is widely used in fields like economics, biology, and earth sciences to understand and demonstrate scaling relationships effectively.

Logarithmic Properties

Understanding logarithmic properties is essential for simplifying and solving exponential and multiplicative relationships. Logarithms essentially "unwrap" these complex relationships, making them easier to work with. Here are some key properties we exploited in the exercise:

• \( \ln(ab) = \ln(a) + \ln(b) \): This property shows how the logarithm of a product can be broken down into the sum of the logarithms.
• \( \ln(b^c) = c \ln(b) \): Logarithms convert exponents into multipliers, simplifying the process of dealing with powers.

In our example, we used these properties to transform the multiplicative relationship in \( R(t) = 3.6 t^{1.2} \) into a sum of logarithms, \( \ln(R(t)) = \ln(3.6) + 1.2 \ln(t) \). By doing this, we turned a power function into a format that makes it easier to interpret and graph as a linear relationship.

Mastery of these logarithmic properties not only helps in solving mathematical exercises but also provides powerful tools in data analysis, allowing complex behaviours to be unraveled and understood.