Problem 65 - 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. $$ y=4 x^{-3} $$

Step-by-Step Solution

Answer

The linear relationship is \( \log(y) = -3\log(x) + \log(4) \), which graphs as a straight line on a log-log plot.

Step 1: Identify the Original Equation

The given equation is \( y = 4x^{-3} \). Our task is to transform this equation to find a linear relationship on a log-log plot.

Step 2: Apply Logarithmic Transformation

Take the logarithm of both sides of the equation: \( \log(y) = \log(4x^{-3}) \). Use the property of logarithms that the log of a product is the sum of the logs: \( \log(y) = \log(4) + \log(x^{-3}) \).

Step 3: Simplify Using Logarithmic Properties

Apply the power rule of logarithms, \( \log(a^b) = b\log(a) \): \( \log(y) = \log(4) - 3\log(x) \). This is of the form \( \log(y) = -3\log(x) + \log(4) \).

Step 4: Express in Linear Form

Rewrite the equation as \( \ log(y) = -3\log(x) + \log(4) \. \) Now, it resembles the equation of a line \( Y = mX + b \), where \( Y = \log(y) \), \( X = \log(x) \), the slope \( m = -3 \), and the y-intercept \( b = \log(4) \).

Step 5: Graph on Log-Log Plot

Plot \( \log(y) \) versus \( \log(x) \). The graph will be a straight line with a slope of -3 and it will intersect the y-axis at \( \log(4) \).

Key Concepts

Logarithmic Properties

Logarithms are incredibly useful in simplifying complex equations, especially those involving products or powers. The key logarithmic properties that often come into play include:

• Product Rule: The logarithm of a product is the sum of the logarithms: \( \log(ab) = \log(a) + \log(b) \).
• Quotient Rule: The logarithm of a quotient is the difference of the logarithms: \( \log(\frac{a}{b}) = \log(a) - \log(b) \).
• Power Rule: The logarithm of a power is the exponent times the logarithm of the base: \( \log(a^b) = b\log(a) \).

These properties simplify expressions and equations, making it easier to find relationships or transformations, as seen when transforming the given equation \( y = 4x^{-3} \) into its linear form.

Linear Relationship

Linear relationships are equations that create straight line graphs. When an equation is transformed into a linear form, like \( \log(y) = -3\log(x) + \log(4) \), it aligns with the standard linear equation format \( Y = mX + b \).
In this transformation:

• \( Y = \log(y) \) and \( X = \log(x) \)
• The slope (\( m \)) is -3, indicating the steepness and direction of the line.
• The y-intercept (\( b \)) is \( \log(4) \), telling where the line crosses the Y-axis when all X values are zero.

Understanding these components is essential for interpreting or predicting behaviors in various datasets. It helps to visualize how changes in one variable affect another in a straightforward, proportional manner.

Graphing on Log-Log Plot

Log-log plots are a brilliant way to visualize data that follow a power-law relationship. In a log-log plot, both the x-axis and y-axis are logarithmic scales.
This plot is particularly effective for graphs where the data points spread across several orders of magnitude, as it aides in seeing straight line patterns within such data. For our transformed equation \( \log(y) = -3\log(x) + \log(4) \):

• The slope of the line on this plot represents the power in the original equation, which is -3 in this instance, making the line decline steeply as x increases.
• The y-intercept \( \log(4) \) helps determine the starting point of the line when \( \log(x) \) is zero.

Drawing and analyzing a log-log plot provides invaluable insight into how quantities scale with one another, especially in scientific and engineering contexts.