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

Step-by-Step Solution

Answer

Taking the logarithm linearizes \( y = x^6 \) to \( \log(y) = 6 \cdot \log(x) \). Plotting this on a log-log scale yields a straight line with a slope of 6.

Step 1: Understand the given equation

We begin with the equation \( y = x^6 \). This equation suggests a power relationship between the variables \( y \) and \( x \), where \( x \) is raised to the sixth power.

Step 2: Apply logarithmic transformation

To linearize the equation, take the logarithm of both sides. This results in: \( \log(y) = \log(x^6) \). Using the logarithmic identity \( \log(a^b) = b \cdot \log(a) \), we can simplify it to \( \log(y) = 6 \cdot \log(x) \).

Step 3: Interpret the linear relationship

The equation \( \log(y) = 6 \cdot \log(x) \) represents a linear relationship \( Y = mX + c \), where \( Y = \log(y) \), \( X = \log(x) \), the slope \( m = 6 \), and the intercept \( c = 0 \).

Step 4: Graph on a log-log plot

On a log-log plot, plot the x-axis as \( \log(x) \) and the y-axis as \( \log(y) \). The resulting graph will be a straight line with a slope of 6, indicating a linear relationship between \( \log(y) \) and \( \log(x) \).

Key Concepts

Power Relationship

A power relationship in mathematics is a type of equation where one variable is proportional to a power of another variable. In the context of our exercise, we have the equation \( y = x^6 \), which is a classic example of a power relationship. Here, the variable \( y \) depends on \( x \) raised to the sixth power.

One of the key characteristics of power relationships is that they can model various natural phenomena and processes. For example, the area of a circle relates to the square of its radius, and the gravitational force between two objects is related to the inverse square of the distance between them.

• These equations can often describe non-linear behavior, meaning that as \( x \) changes, \( y \) does not necessarily change at a constant rate.
• Understanding transformations of power relationships helps in analyzing data that don't initially appear linear.

Recognizing a power relationship allows us to apply a logarithmic transformation, which can convert the equation into a more digestible format, often revealing underlying linear structures.

Linear Relationship

Once a power relationship is present, applying a logarithmic transformation can reveal a linear relationship. In our example, taking the logarithm of both sides of the equation \( y = x^6 \) results in \( \log(y) = 6 \cdot \log(x) \). This conversion rewrites the equation in a linear form.

A linear relationship is defined by the equation \( Y = mX + c \), where \( Y \) and \( X \) are dependent and independent variables, respectively. The term \( m \) represents the slope of the line, indicating how much \( Y \) changes with \( X \).

In the transformed equation, \( \log(y) = 6 \cdot \log(x) \), the slope \( m \) is 6, and the y-intercept \( c \) is 0. This tells us that for every unit increase in \( \log(x) \), \( \log(y) \) increases by 6 units, highlighting a direct linear dependence:

• The slope provides insight into the rate of change between two variables in a log scale.
• A linear model is easier to work with mathematically and provides clearer insights for analysis and prediction.

By interpreting power relationships as linear ones through logarithmic transformations, you can simplify complex datasets and more readily identify trends.

Log-Log Plot

A log-log plot is a graphical representation technique used to visualize data that follow a power law distribution. In this type of plot, both axes are on a logarithmic scale. This method is especially useful for visualizing the linear relationship resulting from the logarithmic transformation of a power equation.

When graphing the transformed equation \( \log(y) = 6 \cdot \log(x) \) on a log-log plot, the x-axis represents \( \log(x) \), and the y-axis represents \( \log(y) \). The resulting graph is a straight line, directly showcasing the linear relationship with a slope of 6.

• A log-log plot is advantageous for revealing proportional disparities across broad data ranges.
• They help in identifying the existence of power laws by transforming multiplicative relationships into additive ones.

This type of plot helps in identifying underlying patterns that are not obvious in standard cartesian plots, making it easier to spot trends or validate theoretical models. It simplifies the interpretation of power relationships, offering clear visual confirmation of linear dependence once a logarithmic scale is applied.