Starting with the first point, (3, 10), we have:
10 = ab^3
And for the second point, (4, 30), we have:
30 = ab^4
We can now use these equations to solve for a and b. One way to do this is to divide the second equation by the first equation:
30/10 = (ab^4)/(ab^3)
Simplifying this expression gives:
3 = b
Substituting this value of b back into one of the original equations (say, the first one) gives:
10 = a(3^3) = 27a
Solving for a gives:
a = 10/27
Therefore, the equation of the exponential function is:
f(x) = (10/27)*(3^x)
Alternatively, we could have used the logarithmic form of the exponential function to solve for a and b:
f(x) = ab^x
log(f(x)) = log(a) + x*log(b)
Letting y = log(f(x)) and u = log(b), we can rewrite the equation in the form of a linear equation:
y = mx + b
where m = u and b = log(a). Then we can use the given points to solve for m and b using standard linear regression techniques. Once we have found m and b, we can substitute them back into the equation to get the final form of the exponential function.