Problem 59 - 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 o g ~ p l o t . ~ $$ y=2 x^{5} $$

Step-by-Step Solution

Answer

Graph \( \ln(y) = 5\ln(x) + \ln(2) \) on a log-log plot for a linear relationship.

Step 1: Identify the Equation

We start with the given equation: \( y = 2x^5 \). The goal is to transform this equation into a form that can be represented as a linear equation using logarithmic transformation.

Step 2: Apply Logarithm to Both Sides

To linearize the equation, apply the natural logarithm \( \ln \) to both sides: \( \ln(y) = \ln(2x^5) \). This uses the property of logarithms that allows the multiplication inside the log to be broken into addition.

Step 3: Use Logarithm Properties

Utilize the properties of logarithms to simplify \( \ln(2x^5) \). We can separate it using: \( \ln(2x^5) = \ln(2) + \ln(x^5) \). Further apply the power rule of logarithms: \( \ln(x^5) = 5\ln(x) \). Thus, \( \ln(y) = \ln(2) + 5\ln(x) \).

Step 4: Represent as a Linear Equation

The transformed equation \( \ln(y) = \ln(2) + 5\ln(x) \) can be equated to the form of a linear equation \( Y = c + mX \), where \( Y = \ln(y) \), \( c = \ln(2) \), \( X = \ln(x) \), and \( m = 5 \).

Step 5: Plot on a Log-Log Graph

Plot \( Y = \ln(y) \) against \( X = \ln(x) \) using a log-log scale. The graph will be a straight line with the slope \( m = 5 \) and the intercept \( c = \ln(2) \). This confirms the linear relationship in a log-log plot.

Key Concepts

Linear Relationship

A linear relationship describes a situation where two variables change in a consistent and predictable manner. In the context of our transformed equation, the relationship between the variables can be described using a straight-line formula: \( Y = c + mX \). Here, \( Y \) and \( X \) represent the transformed logarithmic values of the original variables, and \( c \) and \( m \) are constants which denote the intercept and slope, respectively.

When we apply logarithm transformations to nonlinear equations like \( y = 2x^5 \), our goal is to transform them into a form where the dependent variable can be expressed as a linear combination of independent variable(s).

• The intercept \( c \) is \( \ln(2) \), representing where our line crosses the Y-axis on a log-log plot.
• The slope \( m \) is \( 5 \), indicating how much \( Y \) changes for a unit change in \( X \).

By transforming nonlinear relationships into linear terms, we make it easier to analyze, visualize, and interpret complex data.

Understanding this transformation helps in predicting one variable based on another using simple linear regression methodologies.

Log-Log Plot

A log-log plot is a type of graph used to examine the relationship between two variables when both their scales are logarithmic. Instead of plotting the values of \( y \) and \( x \) directly, a log-log plot represents \( \ln(y) \) against \( \ln(x) \). This is particularly useful for relationships where the data covers several orders of magnitude.

In this plot, the slope and intercept from the linear equation \( \ln(y) = \ln(2) + 5\ln(x) \) become visually clear:

• The slope of the line gives the exponent in the original power-law relationship, which is \( 5 \) in our equation.
• The intercept indicates the multiplicative constant, shown by \( \ln(2) \) in the equation.

Log-log plots are beneficial in fields such as economics, biology, and engineering, where exponential relationships can be revealed as linear.

By understanding the log-log plot, analysts can better identify trends and predict outcomes based on observed data. This visual representation simplifies the interpretation of complex power laws.

Properties of Logarithms

Understanding the properties of logarithms is crucial for simplifying complex expressions and transforming equations like \( y = 2x^5 \) into linear form. Several fundamental properties of logarithms are used to manipulate expressions:

• Product Rule: \( \ln(ab) = \ln(a) + \ln(b) \)
• Power Rule: \( \ln(a^b) = b\ln(a) \)
• Quotient Rule: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)

In our exercise, these properties allow for:
- Breaking down \( \ln(2x^5) \) into \( \ln(2) + 5\ln(x) \), enabling its transformation into a linear equation.

These logarithm properties are central in calculus and algebra, enabling the transformation of exponential data into manageable linear approximations. Knowing how to apply these rules can simplify solving complex equations and understanding datasets that follow exponential trends.

In essence, logarithmic properties bridge the gap between nonlinear and linear perspectives, providing a powerful tool in mathematical analysis.