Suppose you have a variable
F which is a exponential function of a variable
t, for some initial value
P; in other words
. You do not know the values of
r or
P.
You have two observations of
F:
F(5) = 164.8721, and
F(15) = 448.1689. You want to determine
F(10) (i.e., halfway between
t = 5 and
t = 15), but you know that a simple linear interpolation between your two observations won't work, as
F follows an exponential curve.
So hit it this way: Take the natural logs of your two observations...
ln(164.8721) =
5.10517; and ln(448.1689) =
6.10517.
Now linearly interpolate halfway between
these two (5.60517), and exponentiate to get your desired result:
= 271.8281.
When I then reveal that I initially used
P = 100 and
r = 0.10 to generate the numbers, you can verify the two observations, as well as verify that F(10) is indeed 271.8281.
It works because of the following: If you know that
F is a function of
t of the form
, then taking nat logs of both sides...
ln(
F) = ln(
P) +
rt
...shows the nat log of
F as a
linear function of
t, for some constants ln(
P) and
r. Thus, the nat log of
F(10) must certainly fall on the line between the nat logs of your two observations.
Hope that helped out a bit, and good luck!
Linear Mode
