z(5)=9,z(6)=12,z(7)=15,z(8)=18. If z = lny, 
find a formula for y in terms of x.

9, 12, 15, 18 forms an arithmetic sequence 
with common difference = d = 3. So we make  
lists of x, 3x, z(x) and 3x-z(x) 

 x  3x  z(x) 3x-z(x) 
 5  15   9     6
 6  18  12     6
 7  21  15     6
 8  24  18     6

We see that 3x - z(x) = 6 in every case, so

            3x - z(x) = 6
                -z(x) = -3x + 6
                 z(x) = 3x - 6

Since z(x) = ln(y), y = ez(x) = e3x-6

or if you prefer, y = e3(x-2)

Edwin