i don't get what was so hard about this problem. i'm not a smart guy, so that's saying something.

when asked to express something in terms of something else, it means you want to end up with something = something else. so the question asked to express log(base 10) 5 in terms of P and Q, that means we want log(base 10)5 = somefunction of P and Q

i will write log[a]b to mean log to the base a of b.

we have log[8]3 = P

=> 8^P = 3

we have log[3]5 = Q

=> 3^Q = 5

Now since 3 = 8^P we can replace the 3 in the last equation with 8^P, so we have (8^P)^Q = 8^PQ = 5

now take log[10] of both sides, we get

log[10]8^PQ = log[10]5

=> log[10]5 = PQlog[10]8

OR

log[8]3 = P log[3]5 = Q

apply the change of base formula and change both to base 10, we get

log[10]3/log[10]8 = P and log[10]5/log[10]3 = Q

so we have log[10]3 = Plog[10]8 and log[10]3 = log[10]5/Q

equating both these equations we get:

Plog[10]8 = log[10]5/Q ........now multiply both sides by Q

=> PQlog[10]8 = log[10]5

=> log[10]5 = PQlog[10]8

that wasn't that hard, you scared me with the TA thing

did you understand the steps. i did it by two methods, hopefully you get one of them. for the first i used the definition of a logarithm, for the second, i used the change of base formula