Problem 82 - 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. $$ y=3 e^{-2 x} ; \text { base } 3 $$

Step-by-Step Solution

Answer

The linearized equation is \( \log_3(y) = -2x \log_3(e) + 1 \). Graph \( x \) vs. \( \log_3(y) \).

Step 1: Recognizing the Exponential Relationship

We start with the exponential equation given by \( y = 3 e^{-2x} \). Our task is to linearize the relationship by applying a logarithmic transformation with base 3.

Step 2: Applying Logarithmic Transformation

Take the logarithm base 3 of both sides of the equation \( y = 3 e^{-2x} \). This gives \( \log_3(y) = \log_3(3 e^{-2x}) \).

Step 3: Expanding the Logarithmic Equation

Using the properties of logarithms, \( \log_3(3 e^{-2x}) \) becomes \( \log_3(3) + \log_3(e^{-2x}) \). Since \( \log_3(3) = 1 \), the equation simplifies to \( \log_3(y) = 1 + \log_3(e^{-2x}) \).

Step 4: Simplifying the Exponential Transformation

Apply the change of base formula to \( \log_3(e^{-2x}) \). The change of base formula gives us \( \log_3(e^{-2x}) = -2x \log_3(e) \).

Step 5: Forming the Linear Relationship

Substituting our findings from Step 3 and Step 4, the transformed equation is \( \log_3(y) = 1 - 2x \log_3(e) \). This simplifies to the linear equation \( \log_3(y) = -2x \log_3(e) + 1 \).

Step 6: Graphing the Linear Relationship

This equation represents a straight line in the \( \log_3(y) \) vs. \( x \) plane with a slope of \(-2 \log_3(e)\) and a y-intercept of 1. To graph this relationship, plot \( x \) on the x-axis and \( \log_3(y) \) on the y-axis, following the linear equation.

Key Concepts

Exponential Functions

Exponential functions are powerful mathematical models used to represent growth and decay. In our exercise, the function is given by \( y = 3 e^{-2x} \). Here, \( e \) is the base of the natural logarithm and signifies the rapid changes characteristic of exponential behavior.
Understanding exponential functions requires familiarity with their components:

• **Coefficient**: The number 3 in the equation determines the initial value of the function when \( x = 0 \).
• **Base**: In this exercise, the base \( e \) connects to natural processes, commonly appearing in continuous growth or decay.
• **Exponent**: The exponent \(-2x\) signifies the rate of decay. A negative sign implies a decrease as \( x \) increases.

An important characteristic of exponential functions is their constant rate of change, depicted by how quickly they rise or fall as \( x \) varies. This sharp change is what makes them useful in modeling real-world phenomena from populations to physics.

Linearization

Linearization is the process of converting a complex equation, like an exponential function, into a linear form. In simpler terms, it's like straightening a curve.
In mathematical terms, this process makes the relationships easier to visualize and analyze. For example, in our exercise, we used logging to transform the exponential relationship \( y = 3e^{-2x} \) into a linear one.
Here's a breakdown of the linearization steps used:

• **Logarithmic Transformation**: Logarithms are crucial in this process. We applied base 3 logarithms to simplify the exponential, utilizing properties like \( \log(a\cdot b) = \log(a) + \log(b) \).
• **Utilizing Change of Base Formula**: This technique allowed us to express the natural base \( e \) in terms of base 3, and helped simplify the equation further.
• **Achieving a Straight-Line Equation**: After logarithmic transformation, our equation became linear. It looked like \( \log_3(y) = -2x \log_3(e) + 1 \), making it analogous to the formula \( y = mx + b \) of a straight line, with slope \(-2 \log_3(e)\) and intercept 1.

Linearization is key in this context since linear equations are much simpler to work with and their solutions provide more intuitive insights into the behavior of the original function.

Graphing Techniques

Graphing is a fundamental way to visualize mathematical functions. For our transformed linear function, getting the graph correct is essential to understanding the evolution of the relationship between the variables involved.
With our linear equation \( \log_3(y) = -2x \log_3(e) + 1 \), here's how we would craft the graph:

• **Axes Definition**: Plot \( x \) on the horizontal axis and \( \log_3(y) \) on the vertical axis. This setup aligns with the standard form of linear equations.
• **Identify Key Points**: Start by plotting the y-intercept, which is the point where the line crosses the y-axis. For our line, that's at 1 on the \( \log_3(y) \) axis while \( x \) is 0.
• **Utilize the Slope**: The slope \(-2\log_3(e)\) tells us how steep the line is. Move downwards and to the right for positive \( x \). This indicates a decrease in \( y \) as \( x \) increases, due to the negative slope.

Graphical representations of logarithmically transformed functions unlock visual analysis. Seeing the relationship drawn out on a graph provides clarity and concrete understanding of the linear transformation and its implications on the original exponential form.