Problem 61 - 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-log plot. $$ y=x^{6} $$

Step-by-Step Solution

Answer

The linear relationship is \( \ln(y) = 6 \ln(x) \), a line with a slope of 6 on a log-log plot.

Step 1: Understand the Original Relationship

We are given the equation \( y = x^6 \). Our task is to find a linear relationship using a logarithmic transformation.

Step 2: Apply Logarithmic Transformation

We'll apply the logarithm to both sides of the equation. Using base 10 or natural logarithms, we get either \( \log(y) = \log(x^6) \) or \( \ln(y) = \ln(x^6) \). For simplicity, we'll proceed with natural logarithms: \( \ln(y) = \ln(x^6) \).

Step 3: Use the Logarithmic Power Rule

Using the logarithmic power rule \( \ln(x^a) = a \cdot \ln(x) \), we rewrite \( \ln(y) = 6 \ln(x) \). This represents a linear relationship \( \ln(y) = 6 \ln(x) \) where the slope is 6 and there is no intercept.

Step 4: Graph the Linear Relationship

To graph the linear relationship, plot \( \ln(y) \) on the vertical axis and \( \ln(x) \) on the horizontal axis. The resulting graph should be a straight line with a slope of 6, indicating the linear relationship on a log-log plot.

Key Concepts

Linear Relationship

A linear relationship between two variables means that as one variable changes, the other changes in a way that can be represented by a straight line. In math, this is usually expressed by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. However, sometimes relationships aren't initially linear, but we can transform them to become linear. This is useful because linear relationships are simpler to analyze and understand.
Logarithmic transformation is a powerful tool for finding linear relationships from nonlinear data. By applying logarithms to both the dependent and independent variables, we often uncover a linear trend. This transformation helps in making complex equations easier to graph and interpret, especially when dealing with exponential growth or power laws.

Log-Log Plot

A log-log plot is a type of graph used to display data in a way that both the x-axis and y-axis use logarithmic scales. This is particularly useful for visualizing power-law relationships, where one variable is proportional to a power of another.
In a log-log plot, if the original equation is \( y = x^6 \), by taking the logarithms of both sides, we transform \( y \) to \( \ln(y) \) and \( x \) to \( \ln(x) \). The resulting graph of \( \ln(y) = 6 \ln(x) \) will be a straight line, confirming the linearity suggested by the equation.

• The slope of this line is equal to the exponent in the original equation, in this case, 6.
• There is no y-intercept as it goes through the origin in this context.

Log-log plots are a staple in data visualization, making it easier to interpret multiplicative processes or exponential relationships.

Power Rule

The power rule for logarithms is an important concept in algebra and calculus. It states that \( \log_b(x^a) = a \cdot \log_b(x) \), allowing us to take the exponent and multiply it by the log of the base.
This property is a key reason why logarithmic transformations are so useful. For the problem at hand, using the power rule, \( y = x^6 \) transforms to \( \ln(y) = 6 \ln(x) \). This pivotal step turns a complex power equation into a simple linear one.

• This transformation simplifies the process of solving equations analytically.
• It helps reduce computational complexity in graph analysis.
• Overall, it highlights the relationship between multiplicative processes.

Grasping the power rule allows students to manipulate equations efficiently, making it a foundational concept for understanding logarithmic transformations and linear relationships.