Problem 81 - Calculus for Biology and Medicine

Question

Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship in a coordinate system whose axes are accordingly transformed so that a straight line results. $$ y=2^{-x} ; \text { base } 2 $$

Step-by-Step Solution

Answer

The transformed equation is \( \log_2(y) = -x \), which is a line with slope \(-1\) through the origin.

Step 1: Understand the Exponential Equation

The given exponential equation is \( y = 2^{-x} \). This represents an exponential decay, where \( y \) decreases as \( x \) increases. Our task is to transform this equation into a linear form using logarithms.

Step 2: Apply Logarithmic Transformation

We take the logarithm of both sides of the equation with base 2. The transformation is \( \log_2(y) = \log_2(2^{-x}) \). Using the properties of logarithms, particularly \( \log_b(a^c) = c \cdot \log_b(a) \), we simplify the right side: \( \log_2(y) = -x \cdot \log_2(2) \). Since \( \log_2(2) = 1 \), the equation simplifies to \( \log_2(y) = -x \).

Step 3: Rearrange to Linear Form

In the equation \( \log_2(y) = -x \), this can be seen as the equation of a line in the form \( Y = mX + b \), where \( Y = \log_2(y) \), \( m = -1 \), \( X = x \), and \( b = 0 \). This shows a linear relationship where the slope of the line is \(-1\) and the y-intercept is \(0\).

Step 4: Graph the Linear Relationship

With the transformed equation \( \log_2(y) = -x \), plot this in a coordinate system where the vertical axis represents \( \log_2(y) \) and the horizontal axis represents \( x \). The line will slope downward from left to right with a slope of \(-1\), passing through the origin (0,0).

Key Concepts

Exponential Decay

Exponential decay is a mathematical concept where a quantity decreases over time at a rate proportional to its current value. This concept is often characterized by functions of the form \( y = a \, b^{-x} \), where:

• \( y \) is the quantity of interest that changes over time.
• \( a \) is the initial value when \( x = 0 \).
• \( b \) is the base of the exponential function, representing the factor by which the quantity changes at each step.

In the provided exercise, the equation \( y = 2^{-x} \) represents such an exponential decay. This means that as the value of \( x \) increases, the value of \( y \) decreases exponentially. This behavior is quite common in natural processes like radioactive decay, cooling of objects, and depreciation of assets.
Understanding exponential decay helps in predicting how a process will change over time, which is especially useful in fields such as finance, physics, and biology.

Linear Relationship

A linear relationship is one of the simplest forms of relationships in mathematics, described by the equation \( y = mx + c \), where:

• \( y \) and \( x \) are variables representing coordinates in a plane.
• \( m \) is the slope, indicating the steepness and direction of the line.
• \( c \) is the y-intercept, representing the point where the line crosses the y-axis.

In our exercise, after applying a logarithmic transformation to the exponential decay equation, we derived \( \log_2(y) = -x \). This is a special form of a linear equation where \( m = -1 \) and \( c = 0 \).
This means the slope is negative one, indicating a downward-sloping line that passes through the origin. Linear relationships are fundamental because they are predictable and easy to work with, making them an essential tool for analyzing data and understanding basic mathematical principles.

Graphing Techniques

Graphing techniques involve plotting data points on a coordinate plane to visually represent mathematical relationships. With exponential and logarithmic functions, graphing helps us better understand their behavior and application in real-life scenarios.
To graph the result of our logarithmic transformation \( \log_2(y) = -x \), we modify the traditional axes:

• The vertical axis becomes \( \log_2(y) \).
• The horizontal axis remains \( x \).

This transformation is crucial as it allows exponential decay to be displayed as a straight line, highlighting the linearity of the relationship. When graphing:

• Start by plotting the origin at (0,0), as this is the intercept of our line.
• Draw a line with a consistent downward slope to illustrate \( m = -1 \).

These techniques help in visualizing complex mathematical functions and can be further used to interpret and predict trends in various disciplines such as statistics, physics, and economics.