Creating numbers for an exponential growth curve

[Babymech]

I have a question which might be possible to solve entirely through clever
math, but I'm not seeing it yet.

I want to define a LOW number, and a HIGH Number and then, for a given
number of steps, generate numbers in between LOW and HIGH, rising
exponentially from LOW to HIGH. One example would be the following

LOW = 5
HIGH = 100
# of steps: 12

Generated Numbers: 5 5 5 5,5 6 7 8 10 13 18 30 50 100

This, when put into a chart, gives an exponentially rising curve. I know
that I could do this by asking Excel to draw a trend line between a limited
number of points, but I want to generate actual numbers.

My problem so far is that the equations I use to generate the numbers have
to be manually adjusted if I want them to end at up HIGH by, say, the 10th
number. By just inputting LOW, HIGH, and a number representing the
'exponentiality' of the progression, I want the list of numbers to cover the
range between LOW and HIGH in, say, 10 steps, no matter what I input as low
or high values.

Hopefully this makes sense, but I can try to clarify if my meaning doesn't
come across.

Thanks,
Babymech

 

[Bernard Liengme]

We want to model y = A exp(kx)
with x going from 1 to 12
In A1 I entered the LOW value (5)
In A2 the HIGH value (100)
In A3 I enter 1 for the value of k
In A4 I entered 1 fro the value of A
In D1 I entered =$B$4*EXP(ROW(A1)*$B$3)
Copied this dowm to D12
(why do you say "steps 12" but have 14 values?)
Then I had Solver make D1 =A1 and D12 =A2 by varying A3 and A4 (K and A)
Just use two constraints with no target cell
The Solver result was k =0.2723 and A = 3.808


These are the values I get
4.999999991
6.565162161
8.620270855
11.31869523
14.86181395
19.51404375
25.62257238
33.64326859
44.17470287
58.00281765
76.15958088
99.9999999

 

[Bernard Liengme]

Of course, we could do this mathematically
y =AExp(kx)
Ln(y) = Ln(A) + kx

Ln(y2) = Ln(y1) = k(x2 - x1)
k = (Ln(100) - Log(5) ) /11 = 0.272339298

A=y/Exp(kx)
A=100/Exp(0.272339298*12) = 3.807979048


best wishes