Problem 51 - 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. $$ y=5 \times 2^{4 x} $$

Step-by-Step Solution

Answer

The relationship is linear: \( \ln(y) = 4 \ln(2) \cdot x + \ln(5) \).

Step 1: Understand Logarithmic Transformation

To find a linear relationship from the given exponential equation, we apply a logarithmic transformation. This technique uses the properties of logarithms to simplify the equation and reveal a linear form.

Step 2: Apply Natural Logarithm to Both Sides

Start with the equation: \( y = 5 \times 2^{4x} \). Take the natural logarithm of both sides to obtain \( \ln(y) = \ln(5 \times 2^{4x}) \). Using the property \( \ln(ab) = \ln(a) + \ln(b) \), this becomes \( \ln(y) = \ln(5) + \ln(2^{4x}) \).

Step 3: Simplify using Logarithmic Properties

Apply the power rule of logarithms \( \ln(b^m) = m \ln(b) \) to \( \ln(2^{4x}) \), giving \( \ln(2^{4x}) = 4x \ln(2) \). Thus, the equation becomes: \( \ln(y) = \ln(5) + 4x \ln(2) \).

Step 4: Identify Linear Relationship

The equation takes a linear form \( \ln(y) = 4 \ln(2) \cdot x + \ln(5) \). Here, the dependent variable is \( \ln(y) \), the independent variable is \( x \), the slope is \( 4 \ln(2) \), and the intercept is \( \ln(5) \).

Step 5: Graph the Linear Relationship

On a log-linear plot, plot \( \ln(y) \) on the y-axis against \( x \) on the x-axis. The graph will be a straight line with a slope of \( 4 \ln(2) \) and a y-intercept of \( \ln(5) \).

Key Concepts

linear relationship

A linear relationship is a foundational concept where two quantities are directly related in a way that forms a straight line when graphed. In simpler terms, it means if you double one quantity, the other quantity doubles as well, resulting in consistent and predictable outcomes. This concept is often represented by the equation of a line, which can be written in the form: \( y = mx + b \).
When dealing with exponential equations, like in our example \( y = 5 \times 2^{4x} \), turning it into a linear form helps us understand the rate of growth or decay associated with it. Through a logarithmic transformation, we simplify such exponential equations into a linear form, making them easier to handle and interpret.

• In our transformed equation \( \ln(y) = 4 \ln(2) \cdot x + \ln(5) \), \( \ln(y) \) acts as our dependent variable.
• Here, \( x \) is the independent variable, representing the input or cause.
• The slope of the line is \( 4 \ln(2) \), indicating the rate at which \( y \) changes with respect to \( x \).
• The intercept, \( \ln(5) \), tells us where the line crosses the y-axis, which is when \( x \) equals zero.

Breaking down complex relationships into simpler linear ones allows us to view and predict behavior over time effectively. By understanding this basic principle, you can tackle more advanced problems with confidence.

exponential equation

An exponential equation is one where a constant base is raised to a variable's power. It typically leads to rapid increases or decreases. Our example: \( y = 5 \times 2^{4x} \), showcases how changes in \( x \) exponentiate the effect of the base, leading to quicker growth rates.
The beauty of exponential equations lies in their ability to model real-world phenomena, like population growth or radioactive decay, where changes compound over time.

• Exponential equations are often non-linear, making them complex to comprehend without simplification techniques.
• Logarithmic transformations help transform them into a manageable form, revealing a linear relationship hidden within.
• This transformation involves taking the logarithm of both sides, leveraging properties like \( \ln(ab) = \ln(a) + \ln(b) \) to distill the equation to a simpler form.
• In our example, what seems non-linear in its original form becomes linear, simplifying analysis and rendering patterns more obvious.

Through a combination of understanding exponential growth and utilizing logarithmic transformations, you can unravel the underlying simplicity in these equations. This clarity not only aids in solving equations but also enhances your ability to assess and predict exponential phenomena effectively.

graphing techniques

Graphing techniques involve strategies to visually represent equations, revealing patterns and relationships. When we perform a logarithmic transformation on an exponential equation and graph it, we typically use a particular type of plot known as a log-linear plot.
This is what happens in the context of our transformed equation \( \ln(y) = 4 \ln(2) \cdot x + \ln(5) \):

• The log-linear plot has \( x \) on the horizontal axis and \( \ln(y) \) on the vertical axis, illustrating the relationship visually.
• Because \( \ln(y) \) is plotted, it takes advantage of the logarithmic transformation to display what would otherwise be an exponential curve as a straight line.
• The clarity provided by this method makes it easier to identify the slope and intercept quickly.
• A straight line through the points confirms a linear relationship, simplifying the interpretation and projection of future values.

Mastering these graphing techniques provides a significant edge in analyzing complex data datasets or identifying trends. It allows you to not only confirm your algebraic findings but also present your results in a format that's easy to understand and communicate to others.