The relation between species richness and area for a wide variety of…

The correct answer is Option (4) → log S = log C + Z log A

Species richness S depends on the area A. So log S stands alone on the left, and the slope Z multiplies log A.

Option (1) log Z = log C + S log A — Incorrect. S and Z have changed places. Z is the slope, not the quantity being measured.

Option (2) log A = log C + Z log S — Incorrect. A and S have changed places. Area is what changes, and species richness depends on it.

Option (3) log C = log S + Z log A — Incorrect. C is a constant, the Y-intercept. Species richness S must be on the left side.

Option (4) log S = log C + Z log A — Correct. Species richness S is on the left, the slope Z multiplies log A, and log C is the intercept: only this option has every term in its place.

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