Note: I'm assuming log10^x = 2a means log(10^x) = 2a. If that is wrong, then the answer below would be wrong.
Note: log(10) = 1
log(10^x) = 2a
x*log(10) = 2a
x = 2a
a = x/2
log(10^y) = b/2
y*log(10) = b/2
y = b/2
b = 2y
log(10^p) = 3a - 2b
p*log(10) = 3a - 2b
p = 3a - 2b
p = 3(x/2) - 2(2y)
p = (3/2)x - 4y