Problem 50 - 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 i n e a r ~ p l o t . ~ $$ y=1.5 e^{2 x} $$

Step-by-Step Solution

Answer

The linear relationship is \( \ln(y) = 2x + \ln(1.5) \). This can be plotted as a straight line on a log-linear plot.

Step 1: Identify the Transformation

To convert the exponential relationship into a linear one, logarithmic transformation is required. Since the function is exponential in the form of \( y = 1.5 e^{2x} \), we will apply the natural logarithm (base \( e \)) on both sides of the equation.

Step 2: Apply Logarithmic Transformation

Apply the natural logarithm to both sides of the equation:\[\ln(y) = \ln(1.5 e^{2x})\]Using logarithmic properties, such as \( \ln(a \cdot b) = \ln(a) + \ln(b) \) and \( \ln(e^a) = a \), the equation becomes:\[\ln(y) = \ln(1.5) + 2x\]

Step 3: Rearrange to Linear Form

After applying the logarithm, the transformed equation \( \ln(y) = \ln(1.5) + 2x \) resembles the equation of a line \( y = mx + c \), where \( \ln(1.5) \) is the y-intercept \( c \) and \( 2 \) is the slope \( m \).

Step 4: Plot on a Log-Linear Plot

On a log-linear plot, where the y-axis represents \( \ln(y) \) and the x-axis represents \( x \), the linear relationship \( \ln(y) = 2x + \ln(1.5) \) can be graphed as a straight line. The slope of the line will be \( 2 \) and it will intersect the y-axis at \( \ln(1.5) \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a variable appears in the exponent. They have the general form \( y = a \cdot e^{bx} \), where \( a \) and \( b \) are constants, \( e \) is the base of the natural logarithms (approximately 2.718), and \( x \) is the independent variable. Here's what you need to know about exponential functions:

• Growth and Decay: When \( b > 0 \), the function represents exponential growth; when \( b < 0 \), it represents exponential decay.
• Applications: Exponential functions are used in various fields like finance, biology, and physics, such as modeling population growth, radioactive decay, and compound interest.
• Key Characteristics: Exponential functions grow very quickly, and their graphs display a rapid increase or decrease reflected in the curve.

Understanding how these functions behave is crucial as they form the backbone of many real-world models, making it easier to predict and analyze phenomena across various disciplines.

Linear Relationships

Linear relationships are mathematical expressions showing a constant rate of change between two variables, well-illustrated by a straight line when graphed. They have the typical equation: \( y = mx + c \), where:

• \( m \) is the slope of the line, representing the change in \( y \) with respect to \( x \).
• \( c \) is the y-intercept, indicating where the line crosses the y-axis.

Crucial points about linear relationships include:

• Predictability: Thanks to their straightforward nature, they serve as a solid basis for predictions and understanding trends.
• Simplicity: This type of relationship is one of the simplest, making it easily interpretable in analytical contexts.

Transforming non-linear relationships, like exponential functions, into linear ones using logarithmic transformation helps simplify the analysis and graphical representation of data, turning complex, curvilinear graphs into straight lines.

Natural Logarithms

Natural logarithms are logarithms with the base \( e \), where \( e \approx 2.718 \). They are commonly denoted using \( \ln \) rather than the standard \( \log \). These logarithms are particularly useful in calculus and are applicable in various transformations and equations:

• Logarithmic Properties: Key properties include \( \ln(ab) = \ln(a) + \ln(b) \) and \( \ln(e^a) = a \), which simplify the manipulation and solution of exponential equations.
• Link to Exponential Functions: Since natural logarithms are the inverse of exponential functions, they are essential in solving equations involving exponentials and in transforming exponential data into a linear format.
• Common Use Cases: Natural logarithms appear in continuous compound interest calculations, complex exponential growth or decay models, and various differential equations.

By employing natural logarithms, we can effectively linearize exponential functions, making them much easier to work with, especially for analysis and graphical depiction.