Problem 45 - Calculus for Biology and Medicine

Question

When log \(y\) is graphed as a function of \(x, a\) straight line results. Graph straight lines, each given by two points, on a log-linear plot, and determine the functional relationship. (The original \(x-y\) coordinates are given.) $$ \left(x_{1}, y_{1}\right)=(-2,3),\left(x_{2}, y_{2}\right)=(1,1) $$

Step-by-Step Solution

Answer

The relationship is \( y = 10^{(-0.159x + 0.159)} \).

Step 1: Express the Relationship

Given that when \( \log y \) is plotted against \( x \) we obtain a straight line, we know we can express \( \log y \) as a linear relationship \( \log y = mx + c \) where \( m \) is the slope and \( c \) is the y-intercept.

Step 2: Convert Coordinates to Logarithms

Calculate \( \log y_1 \) and \( \log y_2 \). Since \( y_1 = 3 \) and \( y_2 = 1 \): \[ \log y_1 = \log 3 \approx 0.4771 \] \[ \log y_2 = \log 1 = 0 \] This gives the points \( (-2, 0.4771) \) and \( (1, 0) \).

Step 3: Calculate the Slope (m)

Use the slope formula \( m = \frac{\log y_2 - \log y_1}{x_2 - x_1} \): \[ m = \frac{0 - 0.4771}{1 - (-2)} = \frac{-0.4771}{3} \approx -0.159 \]

Step 4: Calculate the Y-Intercept (c)

Using the linear equation with point \( (1, 0) \), substitute the values to solve for \( c \): \[ \log y = mx + c \Rightarrow 0 = -0.159 \times 1 + c \] \[ c = 0.159 \]

Step 5: Formulate the Logarithmic Equation

Insert the values of \( m \) and \( c \) into the linear equation to get the relationship between \( x \) and \( \log y \): \[ \log y = -0.159x + 0.159 \]

Step 6: Find the Exponential Form

Convert the equation from logarithmic form to exponential form: Since \( \log y = -0.159x + 0.159 \), then \[ y = 10^{(-0.159x + 0.159)} \] This gives the original form of the relationship between \( x \) and \( y \).

Key Concepts

Log-Linear Plot

A log-linear plot is a way to graphically represent relationships between two variables where one of the variables is on a logarithmic scale. This is particularly useful when you are dealing with data that spans several orders of magnitude, such as biological systems, where exponential growth is common. In a log-linear plot:

• The x-axis is typically on a normal (linear) scale.
• The y-axis is on a logarithmic scale, displayed as powers of ten.

This type of plot helps in visualizing exponential relationships in a linear manner. If the logarithm of one variable, say log of y, is a linear function of the other variable, x, then the graph will appear as a straight line. This is what we see in the given exercise. We plotted log(y) versus x and the result is a straight line, perfect for showcasing exponential trends in a simple, comprehensible way.

Exponential Functions

Exponential functions are central in modeling growth and decay processes in biology, such as population dynamics or the spread of diseases. An exponential function is of the form:\[ y = a e^{bx} \ or \ y = a \, 10^{bx} \]where:

• \(a\) is the initial value, or y-intercept.
• \(b\) is the growth (or decay) rate.
• \(x\) is the independent variable, often time.

In our original problem, we derived an exponential equation from a log-linear relationship. When you have a straight line in a log-linear plot, you can switch to an exponential form by applying reverse logarithmic operations. This reveals the underlying exponential nature: \(y = 10^{(-0.159x + 0.159)}\). Such equations is how we describe real-world biological phenomena in a mathematical way, giving insight into how variables change at accelerating rates.

Linear Equations

In mathematics, a linear equation describes a line on a two-dimensional surface. When plotted, these lines can be characterized by their slope and intercept according to the formula:\[ y = mx + c \]where:

• \(m\) is the slope of the line, which shows the angle or steepness of the line.
• \(c\) is the y-intercept, the point where the line crosses the y-axis.

In the context of our exercise, determining the linear equation from the log-linear plot was essential in facilitating the transition from a logarithmic to an exponential description. The process involved:

• Calculating the slope using the points \((-2, 0.4771)\) and \((1, 0)\).
• Substituting into the formula to find the complete linear equation: \(\log y = -0.159x + 0.159\).

Linear equations thus serve as a vital bridge in understanding and modeling complex biological interactions using calculus. They provide a simplified pathway from data interpretation to functional understanding.