Some biologists model the number of species S in a fixed area A (such as an island) by the species-area relationship
log S = log c + k log A
where c and k are positive constants that depend on the type of species and habitat.
(a) Solve the equation for S.
S =
(b) Using part (a), if k = 2 and the area is doubled, then by what magnitude is the number of species increased?
If a= bc then log a=logb +logc
if z=c^m
log z=mlogc
a) S=cA^k
b) A=2Ao K=2
S=c(2Ao)^2=c*2^2*Ao^2
S=4cAo^2
therefor the # of species increased by 4.
(a) To solve the equation for S, we need to get rid of the logarithms. By applying the rules of logarithms, we can rewrite the equation as:
log S = log c + k log A
Using the rule that log a + log b = log (a * b), we can rewrite the equation as:
log S = log (c * A^k)
Now, by applying the rule that log a = b is equivalent to a = 10^b, we can solve for S:
S = 10^(log (c * A^k))
Simplifying further:
S = c * A^k
So, S = c * A^k.
(b) If k = 2 and the area is doubled, then the equation becomes:
S = c * (2A)^2
Expanding the equation:
S = c * 4A^2
Comparing this with the original equation S = c * A^k, we can see that the number of species is increased by a magnitude of 4.
(a) To solve the equation for S, we can start by isolating the term "log S" on one side and the other terms on the other side:
log S = log c + k log A
Next, we can rewrite the equation in exponential form to eliminate the logarithms:
S = 10^(log c + k log A)
Using the logarithmic identity log(ab) = log a + log b, we can simplify further:
S = 10^log c * 10^(k log A)
Applying the exponentiation property 10^(k log A) = (10^log A)^k = A^k, we get:
S = c * A^k
Therefore, the equation for S is:
S = c * A^k
(b) Given that k = 2 and the area is doubled, we can substitute these values into the equation from part (a):
S = c * (2A)^2
Simplifying, we have:
S = 4c * A^2
Comparing this equation to the equation in part (a), we see that the number of species S has increased by a factor of 4 (or increased by a magnitude of 4).