Answer:
A generic exponential function can be written as:
f(x) = A*e^(b*x)
Where A and b are real numbers, that we need to find.
We know that the points (3, 6) and (7, 18) are solutions to the equation, then we have that:
A*e^(b*3) = 6
A*e^(b*7) = 18
This is a system of equations, to solve this we can take the quotient between the two equations, so we remove the variable A.
(A*e^(b*7))/(A*e^(b*3)) = 18/6
e^(b*7)/e^(b*3) = 3
e^(b*7 - b*3) = 3
e^(4*b) = 3
Now we can apply the Ln(x) to both sides, because:
Ln(e^y) = y
then:
Ln(e^(4*b)) = Ln(3)
4*b = Ln(3)
b = Ln(3)/4
Then we have:
f(x) = A*e^(Ln(3)/4*x)
And we can use one of the equations to find the value of A. for example:
6 = A*e^(Ln(3)/4*3)
6/e^(Ln(3)/4*3) = 2.632
Then the exponential function is:
f(x) = 2.632*e^(Ln(3)/4*x)
Then we have that:
f(20) = 2.632*e^(Ln(3)/4*20) = 639.576
Rounding to the next integer, we have:
f(20) = 640
