Problem 53 - Calculus for Biology and Medicine

Question

use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-linear plot. $$ y=4 \times 3^{2 x} $$

Step-by-Step Solution

Answer

Transform the equation to \( \ln(y) = \ln(4) + x \ln(9) \) and plot as a straight line on a log-linear graph with \( \ln(y) \) vs. \( x \).

Step 1: Understand the Problem

We need to transform the given equation into a linear form by using logarithms. The equation is \( y = 4 \times 3^{2x} \). This problem involves expressing \( y \) in terms of \( x \) in a way that results in a straight line when plotted on a log-linear graph.

Step 2: Simplify the Exponential Equation

We rewrite the exponential expression by using a logarithmic transformation. The given equation is \( y = 4 \times 3^{2x} \). Notice that \( 3^{2x} \) can be rewritten as \( (3^2)^x = 9^x \). Therefore, the equation becomes \( y = 4 \times 9^x \).

Step 3: Use Logarithms to Linearize

Take the natural logarithm of both sides of the equation: \( \ln(y) = \ln(4 \times 9^x) \). Using the logarithm property \( \ln(a \times b) = \ln(a) + \ln(b) \), this becomes \( \ln(y) = \ln(4) + \ln(9^x) \). Then, simplify \( \ln(9^x) \) to \( x \ln(9) \), resulting in \( \ln(y) = \ln(4) + x \ln(9) \).

Step 4: Identify the Linear Form

The equation \( \ln(y) = \ln(4) + x \ln(9) \) is in the form \( Y = A + Bx \), where \( Y = \ln(y) \), \( A = \ln(4) \), and \( B = \ln(9) \). This is a linear equation with \( Y \) as the dependent variable, \( x \) as the independent variable, \( A \) as the intercept, and \( B \) as the slope.

Step 5: Plot the Linear Relationship

To plot this on a log-linear graph, use \( x \) as the horizontal axis and \( \ln(y) \) as the vertical axis. The intercept \( \ln(4) \) determines where the line crosses the vertical axis, and \( \ln(9) \) is the slope of the line, determining how steep it is. This linear plot will reflect the original exponential growth relationship as a straight line.

Key Concepts

Log-Linear Plot

Understanding how exponential relationships can be visualized is crucial for simplifying complex data. A log-linear plot can help with this. In a log-linear plot, one axis (usually the y-axis) is on a logarithmic scale. This clever adjustment converts an exponential relationship into a linear one, making it easier to interpret.

Here’s why it’s valuable:

• When an exponential relationship like \( y = 4 \times 9^x \) is plotted on a regular linear scale, the graph curves steeply. It can be challenging to analyze or identify patterns in such plots.
• By taking the logarithm of the y-values, you make the steep curve straighten out into a line, simplifying the analysis process.

Practically, this method involves:

• Calculating the natural logarithm (or another logarithmic base) of the y-values.
• Using these transformed values to plot the data against the x-values on a graph.
• The resulting straight line is much easier to work with, allowing for simpler interpretations and predictions.

Overall, log-linear plots reveal underlying linear relationships in exponential data, which are crucial for statistical modeling, economics, biology, and many other fields.

Linear Relationship

The concept of a linear relationship is simple but extremely important in mathematics. A linear relationship describes a straight-line connection between two variables. For instance, in the equation \( \ln(y) = \ln(4) + x \ln(9) \), the linear relationship shows how \( \ln(y) \) changes as \( x \) changes.

Here's why it's significant:

• Linear relationships are easy to interpret. The graph is a straight line, making it simple to predict one variable given the other.
• The equation \( Y = A + Bx \) captures the essence of this relationship, where \( Y \) is the dependent variable, \( x \) is the independent variable, \( A \) is the intercept, and \( B \) is the slope.
• In our case, \( \ln(y) \) is the linear transformation of \( y \), allowing a clear, linear relationship with \( x \).

This transformation makes calculations and predictions more straightforward, as opposed to handling a non-linear, exponential equation directly. Linear relationships are foundational in various analyses where researchers need to understand patterns and make predictions.

Exponential Equation

Exponential equations are a critical part of mathematics and many scientific fields. These equations are characterized by having a variable in the exponent, like \( y = 4 \times 3^{2x} \). Understanding these equations is key because they describe :

• Growth patterns that are not constant but instead accelerate over time, such as populations, radioactive decay, or compound interest.
• The nature of exponential functions means small changes in the input (such as \( x \)) can lead to large changes in the output (\( y \)).

In everyday terms, exponential equations are all about rapid growth or decay. They aren't linear, which means they don't produce straight-line graphs on a typical linear scale. The transformation using logarithms, as shown in our task, simplifies working with exponential data.

This transformation changes the form of the equation to align it more with linear algebra methods, making it manageable for analysis and prediction. Recognizing when data follows an exponential pattern and knowing how to transform it can be very powerful in decision-making, research, and data science.