Problem 79 - 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 to \( \log_2(y) = x \), the graph is linear with slope 1 on semilog axes.

Step 1: Identify the Function and Base

The given exponential function is \( y = 2^x \), and we are asked to use base 2 for the transformation.

Step 2: Apply Logarithm of the Specified Base

Take the logarithm base 2 of both sides of the equation \( y = 2^x \). This gives us \( \log_2(y) = x \cdot \log_2(2) \). Since \( \log_2(2) = 1 \), the equation simplifies to \( \log_2(y) = x \).

Step 3: Resulting Linear Relationship

The transformed equation \( \log_2(y) = x \) is a linear relationship, where \( x \) is the independent variable and \( \log_2(y) \) the dependent variable. It follows the form \( y = mx + c \) with slope \( m = 1 \) and intercept \( c = 0 \).

Step 4: Graphing the Transformed Equation

Plot \( x \) against \( \log_2(y) \) using a semilog graph, where the y-axis is logarithmically scaled. This will result in a straight line with a slope of 1, indicating a linear relationship on the logarithmically transformed coordinates.

Key Concepts

Understanding Exponential Functions

An exponential function is a type of mathematical expression where a constant base is raised to a variable exponent. In our example, the function given is \( y = 2^x \). Here, the base is \( 2 \), and \( x \) is the exponent that varies. Exponential functions are characterized by rapid growth or decay, depending on the context. This rapid change is due to the multiplication of the base by itself as \( x \) changes. Key characteristics of exponential functions:

• They have a constant base; in this example, the base is \( 2 \).
• The exponent is typically the variable, influencing the function's growth.

Exponential functions often appear in scenarios such as population growth, radioactive decay, and finance, where quantities increase or decrease at a consistently rapid rate.

Linear Relationship from Exponential Functions

To find a linear relationship from an exponential function, a logarithmic transformation is used. By applying the appropriate logarithm (in this case, base 2), we convert the exponential equation \( y = 2^x \) into a linear form. The transformed equation is \( \log_2(y) = x \), which mirrors the form of a classic linear equation \( y = mx + c \). Here, the transformation works because the logarithm of the base itself, \( \log_2(2) \), equals 1. So, the transformed equation simplifies to \( x = \log_2(y) \). This equation becomes a straight line with:

• The slope \( m = 1 \)
• An intercept \( c = 0 \)

This neat transformation allows us to analyze exponential data through a more straightforward linear perspective, making it easier to understand and graph.

Graphing Logarithmically Transformed Equations

When graphing an equation that has undergone logarithmic transformation, such as \( \log_2(y) = x \), we typically use a semilog or logarithmic scale. This kind of graphing converts the exponential growth characteristics into a linear format, enabling simpler interpretation. Steps to graph:

• Plot \( x \) on the horizontal axis (standard scale).
• Plot \( \log_2(y) \) on the vertical axis (logarithmic scale).

Using this method, any exponential function converted into a logarithmic form straightens out into a line, which in this example, has a slope of 1. This demonstrates a linear trend and makes visual assessment of changes in data, such as trends over time, much clearer. Such graphing techniques are invaluable in statistics, science, and finance for illustrating growth and decay phenomena.