Problem 52 - 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. $$ v=6 \times 2^{-0.9 x} $$

Step-by-Step Solution

Answer

The equation linearizes to \( \ln(v) = \ln(6) - 0.9 \ln(2) \cdot x \).

Step 1: Understand the given equation

We have the exponential equation given as \( v = 6 \times 2^{-0.9x} \). Our goal is to transform this into a linear equation using logarithmic transformation.

Step 2: Apply the logarithmic transformation

To linearize the equation, we take the logarithm of both sides of the equation. Use the natural logarithm (\( \ln \)) for this purpose: \( \ln(v) = \ln(6 \times 2^{-0.9x}) \).

Step 3: Use logarithm properties

Apply the property of logarithms that states \( \ln(ab) = \ln(a) + \ln(b) \):\[ \ln(v) = \ln(6) + \ln(2^{-0.9x}) \].

Step 4: Simplify using logarithm of powers

Apply the property \( \ln(a^b) = b\ln(a) \) to \( \ln(2^{-0.9x}) \):\[ \ln(v) = \ln(6) - 0.9x \ln(2) \].

Step 5: Identify the linear form

The transformed equation \( \ln(v) = \ln(6) - 0.9 \ln(2) \cdot x \) is now in the linear form of \( y = mx + c \), where \( y = \ln(v) \), the slope \( m = -0.9 \ln(2) \), and the y-intercept \( c = \ln(6) \).

Step 6: Graph the relationship

Plot the graph with \( x \) on the horizontal axis and \( \ln(v) \) on the vertical axis. The graph is a straight line with a negative slope \( -0.9 \ln(2) \) and y-intercept \( \ln(6) \).

Key Concepts

Linearization of Exponential Functions

When dealing with exponential functions, their nature often makes it difficult to directly visualize the relationship between the variables involved, especially as functions grow rapidly. The task of linearization comes into play to simplify this process. Linearization involves transforming an exponential function into a linear one. This makes analysis and graphing much simpler.

Let's consider the equation given: \( v = 6 \times 2^{-0.9x} \). This is an exponential equation where the dependent variable \( v \) grows exponentially with respect to \( x \). To find a linear relationship, we employ logarithmic transformation. Taking the natural logarithm of both sides, we transform the exponential equation into:
\[ \ln(v) = \ln(6) - 0.9x \ln(2) \]
This transformation reveals a straightforward linear form. As a result, the relationship between \( \ln(v) \) and \( x \) aligns with the linear equation format \( y = mx + c \). This allows us to identify \( \ln(v) \) as \( y \), \( -0.9 \ln(2) \) as the slope \( m \), and \( \ln(6) \) as the y-intercept \( c \). The linearization process unlocks a clearer perspective for understanding how changes in \( x \) affect \( v \).

Logarithmic Properties

Logarithmic properties play a crucial role in transforming and simplifying equations. These properties allow us to decompose and manipulate expressions. One key logarithmic property is for multiplication: if \( a \) and \( b \) are positive numbers, then \( \ln(ab) = \ln(a) + \ln(b) \). This property enables the separation of variables, as seen when we transform \( \ln(6 \times 2^{-0.9x}) \) into \( \ln(6) + \ln(2^{-0.9x}) \).

Another invaluable property is related to powers: for any positive number \( a \) and real number \( b \), \( \ln(a^b) = b \ln(a) \). This allows us to bring down the exponent as a multiplicative factor. Applying this to \( \ln(2^{-0.9x}) \), we simplify it to \( -0.9x \ln(2) \).

These transformations move our complex exponential equation into a more tangible linear form. Understanding and utilizing these logarithmic properties are essential for solving problems in not just mathematics, but also in areas like physics and economics, where exponential growth and decay are prevalent.

Log-Linear Plot

A log-linear plot is a powerful tool in data analysis for visualizing relationships that are not immediately linear. Once linearization of an exponential function is achieved, the next step is to graph the transformed variables. In this scenario, rather than plotting \( v \) directly against \( x \), we plot \( \ln(v) \) versus \( x \).

The result is a graph that displays a straight line when managed correctly using our transformed equation \( \ln(v) = \ln(6) - 0.9x \ln(2) \). Provided \( y = \ln(v) \) and \( x \) is unchanged, this plot's slope corresponds to the coefficient of \( x \), specifically \(-0.9 \ln(2) \), and the y-intercept is \( \ln(6) \).

This kind of plot is beneficial because it enables clear identification of linearity in data. If the log-linear plot results in a straight line, it confirms the initial exponential relationship between \( x \) and \( v \). It's particularly useful in fields like biology and finance, where such patterns frequently occur. Crafting and interpreting log-linear plots help bring clarity to exponential trends and can significantly simplify the analysis.