Question

Identify the "a" value and "b" value from the exponential equation you would get from the points (1, 10.5) and (4, 283.5)?

Answer 1

`f(x) = ae^(bx)" "` where `\ a=7/2 \` and `\ b= ln 3`

or if you prefer: `f(x) = 7/2 3^x`

Explanation:

An exponential function can be described by the formula:

`f(x) = ae^(bx)`

for some constants `a, b` to be determined.

So from the two given points, we have:

`{ (10.5 = ae^b), (283.5 = ae^(4b)) :}`

So:

`3^3 = 27 = 283.5/10.5 = (ae^(4b))/(ae^b) = (e^b)^3`

Assuming we are only interested in real-valued functions of real numbers, this means that:

`3 = e^b`

and hence `b = ln 3`

Then:

`a = 10.5/e^b = 10.5/3 = 7/2`

So `a=7/2`, `b=ln 3` and:

`f(x) = 7/2 e^(x ln 3)`

If you prefer, this is simply:

`f(x) = 7/2 3^x`