Problem 69 - 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. $$ N(t)=130 \times 2^{1.2 t} $$

Step-by-Step Solution

Answer

Use a log-linear plot for graphing.

Step 1: Understand the Expression

We start with the expression \( N(t) = 130 \times 2^{1.2t} \). This represents an exponential growth function where \( N(t) \) is dependent on \( t \). Our goal is to use a logarithmic transformation to linearize this relationship.

Step 2: Take the Logarithm of Both Sides

Apply the natural logarithm to both sides of the equation. We get: \( \ln(N(t)) = \ln(130 \times 2^{1.2t}) \). Using the property of logarithms that \( \ln(ab) = \ln(a) + \ln(b) \), this becomes \( \ln(N(t)) = \ln(130) + \ln(2^{1.2t}) \).

Step 3: Simplify the Logarithmic Expression

Use the property of logarithms that \( \ln(a^b) = b \cdot \ln(a) \) to further simplify: \( \ln(N(t)) = \ln(130) + 1.2t \cdot \ln(2) \). This expression is now linear in terms of \( t \) with \( \ln(N(t)) \) as our dependent variable.

Step 4: Identify the Type of Plot

We have transformed our equation into a linear relationship: \( \ln(N(t)) = \ln(130) + 1.2 \ln(2)\cdot t \). This indicates a log-linear relationship because the logarithm was applied only to \( N(t) \), not to \( t \). As a result, the appropriate type of plot is a semi-log plot, where the y-axis is logarithmic.

Key Concepts

Exponential Growth

In the context of the given function, exponential growth refers to a situation where the quantity, such as a population or investment, increases at a rate proportional to its current size. In the function, \( N(t) = 130 \times 2^{1.2t} \), the number \( N(t) \) grows exponentially over time \( t \). Features of exponential growth:

• Rapid increase: The larger the quantity becomes, the faster it grows.
• Constant percentage increase: The rate of growth is constant in terms of percentage.
• Expressed often in the form of \( N(t) = N_0 \times a^{kt} \), where \( a \) (the base) is greater than 1.

In real life, this kind of growth can be observed in populations, investments with compounded interest, or bacteria cultures in optimal conditions.

Linear Relationship

A linear relationship describes a direct proportionality between two variables. In mathematical terms, a linear equation can be represented as \( y = mx + c \), where \( y \) is the dependent variable, \( x \) is the independent variable, \( m \) is the slope, and \( c \) is the y-intercept. Linear relationships form a straight line when graphed.Key characteristics:

• Constant rate: The rate of change between variables remains constant.
• Directly proportional: Doubling one variable results in doubling the other.
• Simplicity: These relationships are easy to model and predict due to their straightforward nature.

In our exercise, the use of logarithmic transformation helped identify a linear relationship between variables by simplifying the exponential growth into a linear form, \( \ln(N(t)) = \ln(130) + 1.2 \ln(2)\cdot t \).

Semi-log Plot

A semi-log plot is a type of graph that uses a logarithmic scale for one axis, and a linear scale for the other. This is particularly useful when you want to display exponential relationships in a linear format. In the given example, \( N(t) = 130 \times 2^{1.2t} \) transforms to \( \ln(N(t)) = \ln(130) + 1.2 \ln(2)\cdot t \), indicating that a semi-log plot is appropriate.Advantages of a semi-log plot:

• Straight line depiction: Transforms exponential relationships into a straight line, making trends easier to analyze.
• Reduces skew: Useful for data that span several orders of magnitude.
• Clarity: Helps in visualizing changes in rates over time.

By plotting \( \ln(N(t)) \) on the y-axis and \( t \) on the x-axis, you can readily observe a straight line if the relationship is truly exponential.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a logarithm to the base \( e \), where \( e \approx 2.71828 \). The natural logarithm is commonly used in mathematics to solve exponential and logarithmic equations due to its properties, especially when dealing with growth processes.Primary features of natural logarithms:

• Ease of differentiation: \( \frac{d}{dx} \ln(x) = \frac{1}{x} \)
• Integration simplicity: \( \int \frac{1}{x} \, dx = \ln|x| + C \)
• Simplifies power expressions: \( \ln(a^b) = b \ln(a) \)

In the exercise, applying the natural logarithm to the equation \( N(t) = 130 \times 2^{1.2t} \) was essential for linearizing the data, facilitating the identification of a log-linear relationship.