Problem 66 - 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. $$ y=7 x^{-5} $$

Step-by-Step Solution

Answer

The linear relationship after logarithmic transformation is \( \ln(y) = \ln(7) - 5\ln(x) \).

Step 1: Understand the Equation

We have the equation \( y = 7x^{-5} \). This is a power equation, where the output \( y \) is proportional to \( x \) raised to the power of \(-5\). Our goal is to transform this into a linear form by using logarithms.

Step 2: Apply Logarithmic Transformation

Apply the natural logarithm to both sides of the equation to convert the power relationship into a linear relationship:\[\ln(y) = \ln(7x^{-5})\]Using the property of logarithms \(\ln(ab) = \ln(a) + \ln(b)\), this becomes:\[\ln(y) = \ln(7) + \ln(x^{-5})\]

Step 3: Simplify Using Logarithm Properties

Using the logarithmic property \(\ln(x^a) = a\ln(x)\), simplify further:\[\ln(y) = \ln(7) - 5\ln(x)\]This equation is in the form \( Y = A + BX \), where \( Y = \ln(y) \), \( A = \ln(7) \), \( B = -5 \), and \( X = \ln(x) \).

Step 4: Identify Linear Relationship

The above transformation reveals the linear relationship:\[Y = B\ln(x) + A\]This is a straight line with slope \( B = -5 \) and intercept \( A = \ln(7) \).

Step 5: Graph on a Log-Log Plot

To graph this, plot \( \ln(y) \) versus \( \ln(x) \). Because the plot is on log-log scales, the transformation ensures that the original power law relationship appears as a straight line with the specified slope (-5).

Key Concepts

Linear Relationship

A linear relationship is one where two quantities, when plotted against each other, form a straight line. In mathematical terms, this can be represented as: \[ Y = mX + c \] where \( Y \) and \( X \) are variables, \( m \) is the slope, and \( c \) is the y-intercept. The hallmark of a linear relationship is that changes in \( X \) result in proportional changes in \( Y \). This means, for every unit increase in \( X \), \( Y \) will increase (or decrease) by a constant amount, \( m \). In the exercise, our challenge was to transform a power equation into this linear form using a logarithmic transformation: - We took the natural logarithm of both sides of the equation. - Logarithm properties helped break down the components, making it easier to identify the linear form. Through this transformation, the complex behavior of the original equation simplifies, revealing a clear, predictable linear relationship by plotting \( \ln(y) \) against \( \ln(x) \).

Log-Log Plot

A log-log plot is a type of graph used to easily identify power-law relationships by plotting logarithms of variables instead of the variables themselves. This type of plot is particularly useful in scientific data where the relation between quantities follows a power rule, like \( y = 7x^{-5} \). Here's how it works: - By plotting log values, power laws manifest as straight lines. - The slope of that line represents the exponent from the power equation. - It simplifies complex relationships into linear ones, enabling easier analysis.In our exercise solution: - By plotting \( \ln(y) \) against \( \ln(x) \), a clear straight line with a negative slope of -5 emerges on the log-log plot. - This confirms that the original equation behaved according to a power rule, providing a visual representation of its linearity after transformation.

Power Equation

A power equation is characterized by a form similar to \( y = cx^a \), where \( y \) and \( x \) are variables, \( c \) is a constant, and \( a \) is the power exponent. This non-linear relationship means that changes in \( x \) cause y to change according to the power of \( a \). Let's break down its features: - Power equations denote scaling and growth patterns, common in nature and economics.- They can describe phenomena where growth is multiplicative rather than additive.In our given exercise, the power equation \( y=7x^{-5} \) indicates: - \( x \) inversely affects \( y \) as indicated by the negative exponent. - Every sub-step of transforming this equation into a linear form involved understanding the impact of \( x \) raised to \(-5\), and the constant multiplier \( 7 \).Such equations might initially seem intricate, but as we demonstrated, they can often be unraveled into simpler linear forms through logarithmic transformation, facilitating easier interpretation and graphing.