Question

Consider the graph shown, where \(S\) is species richness and \(A\) is area. \(S\) and \(A\) are log-transformed and the slope is not equal to 1.
\includegraphics[width=0.5\linewidth]{46.png}
The relationship between untransformed \(S\) and \(A\) follows a/an:

• A. linear relationship.
• B. power law.
• C. exponential relationship.
• D. Michaelis-Menten function.

Hint:

To analyze log-log graphs: 1. A straight line on a log-log scale indicates a power law relationship.
2. The equation \(S = k A^m\) is derived from \(\log(S) = c + m \log(A)\).
3. Recognize the difference between linear, exponential, and power law relationships based on scaling behavior.

Answer 1

The Correct Option is B

Step 1: Understand the graph. The graph represents the relationship between \(\log(S)\) and \(\log(A)\), with a straight line indicating that the relationship between \(S\) and \(A\) on a logarithmic scale is linear. The equation for such a relationship is: \[ \log(S) = c + m \log(A), \] where \(c\) is the intercept, and \(m\) (the slope) is not equal to 1. Step 2: Convert to the untransformed relationship. Rewriting the equation in its untransformed form: \[ S = k A^m, \] where \(k = 10^c\). This equation describes a power law relationship between \(S\) (species richness) and \(A\) (area), where \(m\) determines the scaling. Step 3: Evaluate the options. Option (A): Incorrect. A linear relationship implies \(S \propto A\), which is not consistent with the power law form \(S = k A^m\). Option (B): Correct. The equation \(S = k A^m\) matches the definition of a power law. Option (C): Incorrect. An exponential relationship would be of the form \(S = k e^{mA}\), which is not implied here. Option (D): Incorrect. A Michaelis-Menten function describes saturation dynamics, not a simple power law.