Which of the following correctly expresses the equation for species-…

The correct answer is Option (2) → \( \log S=\log C+Z \log A \)

Species richness rises with the area explored. On a log scale the relation is a straight line: the log of species richness (S) equals the log of the intercept (C) plus Z times the log of the area (A), Z being the slope.

Option (1) \( \log S=\log C/A \log Z \) — Incorrect. It divides by A and puts Z inside the log; a straight-line form has the slope multiplying log A.

Option (2) \( \log S=\log C+Z \log A \) — Correct. The relationship S = CA^Z is a rectangular hyperbola; on a log scale it becomes the straight line log S = log C + Z log A, where Z is the slope and C the Y-intercept.

Option (3) \( \log S=\log C-Z \log A \) — Incorrect. A minus sign would mean fewer species in a larger area, the opposite of what is observed.

Option (4) \( \log S=Z \log C-\log (A)^2 \) — Incorrect. It puts the slope on C and subtracts the area term; neither fits the relation.

NCERT Extract:
Patterns of Biodiversity – “… (ii) Species-Area relationships : During his pioneering and extensive explorations in the wilderness of South American jungles, the great German naturalist and geographer Alexander von Humboldt observed that within a region species richness increased with increasing explored area, but only up to a limit. In fact, the relation between species richness and area for a wide variety of taxa (angiosperm plants, birds, bats, freshwater fishes) turns out to be a rectangular hyperbola (Figure13.2). On a logarithmic scale, the relationship is a straight line described by the equation
log S = log C + Z log A
where
S= Species richness   A= Area
Z = slope of the line (regression coefficient)
C = Y-intercept
Ecologists have discovered that the value of Z lies in the range of 0.1 to 0.2, regardless of the taxonomic group or the region (whether it is the plants in Britain, birds in California or molluscs in New York state, the slopes of the regression line are amazingly similar).”

Figure 13.2 Showing species area relationship. Note that on log scale the relationship becomes linear