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:
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:
\includegraphics[width=0.5\linewidth]{46.png}
The relationship between untransformed \(S\) and \(A\) follows a/an:
Show 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.
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.
Updated On: Jan 24, 2025
- linear relationship.
- power law.
- exponential relationship.
- Michaelis-Menten function.
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The Correct Option is B
Solution and Explanation
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.
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