To write an exponential function in the form of y = ab^x that passes through two given points (x1, y1) and (x2, y2), we need to first find the values of a and b.
Using the point (2, 16), we get:
16 = ab^2
Using the point (5, 128), we get:
128 = ab^5
We can divide the second equation by the first equation to eliminate a:
(128/16) = (ab^5)/(ab^2)
8 = b^3
Taking the cube root of both sides gives:
b = 2
Now we can substitute this value of b into either equation to solve for a. Using the first equation:
16 = ab^2
16 = a(2^2)
16 = 4a
a = 4
Therefore, the exponential function that passes through the points (2, 16) and (5, 128) is:
y = 4(2^x)