Question 1170843: 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?
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Part (a)
We'll use these log rules
log(A)+log(B) = log(A*B)
B*log(A) = log(A^B)
which we'll refer to as rule 1 and rule 2 respectively.
log(S) = log(c) + k*log(A)
log(S) = log(c) + log(A^k) .... use rule 2
log(S) = log(c*A^k) .... use rule 1
Since the log on the left equals the log on the right, this means the arguments (ie the stuff inside) for each log function must be equal.
In other words, log(A) = log(B) implies that A = B.
The equation
log(S) = log(c*A^k)
turns into
S = c*A^k
Answer: S = c*A^k
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Part (b)
The area is doubled. So A turns into 2A.
We'll replace A with 2A like so
S = c*A^k
S = c*(2A)^k
Now plug in k = 2
S = c*(2A)^k
S = c*(2A)^2
S = c*(4A^2)
Plug k = 2 into the original equation
S = c*A^k
S = c*A^2
Divide the two results
[ c*(4A^2) ] / [ c*A^2 ]
(4A^2)/(A^2)
4
The 'c's cancel and the 'A^2's cancel.
For k = 2, if we double the area from A to 2A, then the number of species has increased by a factor of 4. It turns out this works for any value of k as well.
Answer: 4
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