Problem 52 - 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 . ~ $$ y=3 \times 5^{-1.3 x} $$

Step-by-Step Solution

Answer

The linear relationship is \( \ln(y) = \ln(3) - 1.3x \cdot \ln(5) \) and can be graphed as a straight line with slope \(-1.3\ln(5)\) and y-intercept \(\ln(3)\) on a log-linear plot.

Step 1: Take the Logarithm

To find a linear relationship, we first take the natural logarithm of both sides of the equation. Start with the equation: \( y = 3 \times 5^{-1.3x} \). Applying the logarithm gives: \( \ln(y) = \ln(3 \times 5^{-1.3x}) \).

Step 2: Use Logarithm Properties

Utilize the property that \( \ln(a \times b) = \ln(a) + \ln(b) \). This simplifies our equation to: \( \ln(y) = \ln(3) + \ln(5^{-1.3x}) \).

Step 3: Simplify the Exponent

Apply another property of logarithms, \( \ln(b^c) = c \cdot \ln(b) \), to simplify \( \ln(5^{-1.3x}) \) to \(-1.3x \cdot \ln(5) \). The equation becomes: \( \ln(y) = \ln(3) - 1.3x \cdot \ln(5) \).

Step 4: Rearrange to Linear Form

The equation \( \ln(y) = \ln(3) - 1.3x \cdot \ln(5) \) is in the form of a linear equation \( y = mx + c \), where \( y \) is \( \ln(y) \), \( x \) remains as \( x \), the slope \( m \) is \(-1.3\ln(5)\), and the y-intercept \( c \) is \( \ln(3) \).

Step 5: Graph the Linear Relationship

Plot \( \ln(y) \) against \( x \) on a graph. The slope of the line will be \(-1.3 \ln(5)\) and the y-intercept will be \( \ln(3) \). This results in a straight line in a log-linear plot.

Key Concepts

Linear Relationship

A linear relationship is a type of relationship between two variables where a change in one variable leads to a proportional change in the other. This can be visually represented by a straight line on a graph. So, when we talk about finding a linear relationship, we're looking for a way to express the connection between two variables with a line equation, like \( y = mx + b \).
In order to transform the given equation \( y = 3 \times 5^{-1.3x} \) into a linear form, we need to take advantage of logarithmic transformation. This involves using properties of logarithms to rearrange and simplify the equation until it resembles a linear equation.

• After the logarithmic transformation, our equation looks like \( \ln(y) = \ln(3) - 1.3x \cdot \ln(5) \).
• This expression is now in the familiar linear format \( y = mx + c \), where \( \ln(y) \) plays the role of \( y \), \( x \) remains \( x \), \( m \) is \(-1.3 \ln(5)\), and \( c \) is \( \ln(3) \).

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm that has the base of the mathematical constant \( e \), which is approximately equal to 2.71828. The natural logarithm is widely used in mathematics and statistics because it simplifies many problems involving exponential growth or decay.
In the context of our exercise, we use the natural logarithm to linearize an otherwise non-linear equation. By applying \( \ln \) to both sides of the original equation, we bring down exponents and transform products into sums, which aligns with linear relationship properties.

• The property \( \ln(a \times b) = \ln(a) + \ln(b) \) was used to separate components of the equation.
• The property \( \ln(b^c) = c \cdot \ln(b) \) was utilized to simplify expressions with exponents.

Using these properties, the complex multiplicative and exponential nature of the original equation can be broken down into a linear form that is easy to interpret and plot.
Understanding and applying natural logarithms is essential for determining linear relationships in a variety of fields, from physics to finance.

Log-Linear Plot

A log-linear plot is a graph where one axis (typically the vertical axis) is scaled logarithmically, and the other axis (typically the horizontal axis) remains on a linear scale. This type of plot is very useful for visualizing exponential relationships as straight lines, making it easier to understand and analyze.
In our exercise, we plot \( \ln(y) \) against \( x \) on a log-linear plot. This results in a straight line because of the linearity we achieved after the logarithmic transformation.

• The slope of the line is \(-1.3 \ln(5)\), indicating how quickly \( \ln(y) \) decreases as \( x \) increases.
• The y-intercept is \( \ln(3) \), the point where the line crosses the vertical axis when \( x = 0 \).

Such plots are invaluable for revealing linear trends in data that are initially non-linear. They allow us to see the 'big picture' of data trends clearly and make predictions about future data points. Log-linear plots are frequently used in many scientific fields, such as biology, engineering, and economics to model growth or decay processes.