Problem 54 - 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 . ~ $$ y=2^{x+1} $$

Step-by-Step Solution

Answer

By taking \( \log(y) \), the equation \( \log(y) = x \log(2) + \log(2) \) shows a linear relationship with slope \( \log(2) \).

Step 1: Understand the Given Equation

The given equation is an exponential form where \( y = 2^{x+1} \). Our task is to find a linear relationship by applying a logarithmic transformation.

Step 2: Apply Logarithmic Transformation

Take the logarithm of both sides of the equation. Using the property \( \log(a^b) = b\log(a) \), we apply log base 10 or natural logarithm:\[ \log(y) = \log(2^{x+1}) \]\[ \log(y) = (x+1)\log(2) \]This transformation results in \( \log(y) \) related linearly to \( x \) by the equation \( \log(y) = x\log(2) + \log(2) \).

Step 3: Interpret the Linear Equation

The equation \( \log(y) = x \log(2) + \log(2) \) is in the form of \( y = mx + c \), where:- \( m = \log(2) \) is the slope.- \( c = \log(2) \) is the y-intercept. Thus, the linear relationship is a straight line on a log-linear plot.

Step 4: Set Up the Log-Linear Plot

In a log-linear plot, \( x \) is plotted on the horizontal axis and \( \log(y) \) is plotted on the vertical axis. The result is a straight line with slope \( \log(2) \).

Step 5: Graph the Relationship

To graph, plot points for various values of \( x \) and calculate corresponding \( \log(y) \) using the equation \( \log(y) = x \log(2) + \log(2) \). By joining these points, you'll get a straight line illustrating the linear relationship.

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In the equation provided, \( y = 2^{x+1} \), the base is 2, and the exponent is \( x+1 \). This means that as \( x \) changes, \( y \) will grow or shrink exponentially.
Exponential functions are essential in various fields because they describe processes with rapid growth or decay, such as population growth, radioactive decay, and interest calculations.

• The constant base (here, 2) determines the rate of growth or decay.
• The variable in the exponent signifies the rapid change across different values.

Understanding the behavior of exponential functions is crucial for interpreting how dramatic increases or decreases occur in numerous real-world scenarios.

Linear Relationship

A linear relationship is characterized by a straight line when plotted on a graph. It follows the equation \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. In our transformed equation \( \log(y) = x \log(2) + \log(2) \), we have established a linear relationship between \( x \) and \( \log(y) \).
Instead of dealing directly with the curves typical for exponential functions, transforming them using logarithms simplifies these into linear equations.

• The slope \( m = \log(2) \) shows how much \( \log(y) \) increases when \( x \) increases by one unit.
• The y-intercept \( c = \log(2) \) indicates the value of \( \log(y) \) when \( x = 0 \).

This transformation makes complex exponential relationships easier to predict and analyze, offering significant practical benefits.

Log-Linear Plot

Log-linear plots are used to display exponential data in a linear form. By plotting the logarithm of the data versus the variable, we make the exponentials appear as straight lines on the graph. This can be crucial for visual data evaluation because it simplifies the trends.
In our problem, after transforming \( y = 2^{x+1} \) to \( \log(y) = x \log(2) + \log(2) \), we plot \( x \) on the x-axis and \( \log(y) \) on the y-axis.

• This results in a straight line since the exponential growth is now linear regarding its logarithmic value.
• Such plots are extensively used in scientific data analysis for model validation and prediction.

Log-linear plots help in simplifying the comprehension of the exponential data patterns, thereby making data analysis more approachable.

Logarithm Properties

Logarithms have several fascinating properties that make them especially useful in mathematics for transforming non-linear data into linear forms. The key properties include:

• \( \log(a \times b) = \log(a) + \log(b) \), which allows multiplication to be converted into addition.
• \( \log(a^b) = b\log(a) \), converting powers into multiplication.
• \( \log(1/a) = -\log(a) \), handling inverses by representing them as negatives.

In our context, we used the property \( \log(a^b) = b\log(a) \) to simplify the transformation of the exponential function into a linear form with \( \log(y) = (x+1) \log(2) \).
By understanding these properties, you can manipulate and re-express equations flexibly, making logarithms invaluable in tackling complex problems spanning scientific, financial, and statistical domains.