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 either in log or semilog transformed coordinates so that a straight line results. $$ y=2^{-x} ; \text { base } 2 $$

Step-by-Step Solution

Answer

Transforming with base 2 gives \( \log_2(y) = -x \), a linear relationship with slope \(-1\).

Step 1: Identify the Transformation

The given equation is the exponential relationship \( y = 2^{-x} \). We need to find a logarithmic transformation using base 2 that results in a linear relationship.

Step 2: Apply Logarithm to Both Sides

Apply the logarithm with base 2 to both sides of the equation: \[ \log_2(y) = \log_2(2^{-x}) \].

Step 3: Simplify Using Logarithm Rules

Use the logarithmic identity \( \log_b(a^c) = c \cdot \log_b(a) \). Thus, \[ \log_2(y) = -x \times \log_2(2) \]. Since \( \log_2(2) = 1 \), this simplifies to \( \log_2(y) = -x \).

Step 4: Identify the Linear Relationship

From \( \log_2(y) = -x \), we identify the linear relationship where \( x \) is the independent variable and \( \log_2(y) \) is the dependent variable. The slope is \(-1\), and there is no y-intercept (i.e., the line passes through the origin).

Step 5: Graph the Transformed Equation

To graph the transformed relationship, plot \( x \) on the horizontal axis and \( \log_2(y) \) on the vertical axis. The graph will be a straight line with a slope of \(-1\), showing a direct linear relationship.

Key Concepts

Exponential Relationship

An exponential relationship describes a situation where a quantity grows or decays at a rate proportional to its current value. In mathematical terms, this means that one variable can be expressed using an exponent. To illustrate, let's consider the exponential relationship given by the equation \( y = 2^{-x} \). Here, the output \( y \) decreases exponentially as the input \( x \) increases. This is a typical exponential decay pattern.

The base of the exponential expression is fundamental to understanding its behavior. In the equation \( y = 2^{-x} \), the base is 2. This means that for each increase in \( x \), the value of \( y \) is halved, indicating a rapid decrease. Such relationships are common in real-world scenarios such as radioactive decay, population decline in a constrained environment, or the cooling of hot objects.

Linear Relationship

A linear relationship is a straightforward association between variables where the change in one variable leads to a proportional change in the other. Graphically, it is represented as a straight line. In the context of our original problem, the transformation of the exponential relationship \( y = 2^{-x} \) into a linear form is achieved using logarithms.

After applying the logarithm base 2 to both sides of the equation, we get \( \log_2(y) = -x \). This equation indicates a linear relationship between \( \log_2(y) \) and \( x \), with a slope of \(-1\). The absence of a y-intercept implies the line passes through the origin. The transformation to a linear form makes the relationship easier to analyze and interpret, particularly in statistical and engineering contexts. It allows for straightforward predictions and understanding of the behavior of the variables involved.

Logarithm Rules

Understanding logarithm rules is essential for manipulating and simplifying expressions involving exponents. The logarithm rules are a collection of properties that allow us to transform exponential expressions into linear ones. In our example, the rule \( \log_b(a^c) = c \cdot \log_b(a) \) is pivotal. It states that the logarithm of a number raised to an exponent equals the exponent times the logarithm of the base number.

For the equation \( \log_2(y) = \log_2(2^{-x}) \), we apply this rule to simplify the expression to \( \log_2(y) = -x \times \log_2(2) \). Knowing that \( \log_2(2) = 1 \), we further simplify to \( \log_2(y) = -x \).
This transformation utilizes basic logarithm properties to convert an exponential relationship into a linear one, thus making it more interpretable and more accessible for further analysis. Knowing these rules can help simplify seemingly complex exponential equations into more manageable linear forms.