passing through (1, 180) and (3, 20).
(Note: All of the 'log' in the following are natural logs.)
log(y) = log(c) + k log(x)
Log(180) = log(c) + k log(1) = log(c)
c = 180
log(20) = log(180) + k log(3)
log(20) - log(180) = k log(3)
k = log(20/180) / log(3) = - log(9) / log(3) = -2
y = 180 e-2xIn checking my work, I find that this doesn't work at all.
Oh woe! Where have I failed ?
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Here's what I did:
y=Cekx which passes through the points (1,180) & (3,20)
(1) Substitute the values of x & y to create two equations:
180 = Cek & 20 = Ce3k
(2) Rearrange in terms of the constant, C:
C=180/ek & C=20/e3k
(3) Since both sets of points satisfy the exponential function, the constant will be the same so:
180/ek = 20/e3k
(4) Now rearrange and solve for k:
180e3k = 20ek
9 = e(k-3k)
ln(9) = ln(e-2k)
ln(9) = -2k
k = -ln(9)/2 which is approx. -1.09861
(5) Now substitute k into one of the equations to solve for C:
C =180/e(-1.09861)
C = 539.9987 or rounded to 540
So the exponential function that includeds the points (1,180) & (3,20) can be approximated as:
y = 540ekx , where k = -1.09861
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