Question

Given that log200=0.3010 and log1000=0.4771 calculate without using table log0.72.

Answer 1

first, I think you mean to say that

log 2 = 0.3010
log 3 = 0.4771

Now: 72 = 2^3 * 3^2
so
log 0.72 = 3log2 + 2log3 - 2

Answer 2

You are starting off with a wrong premise
log 200 = 2.301, not as given
log 1000 = 3, not as given

If not stated, log x is assumed to have base 10
If you are working with some mysterious unknown base, then
log_a 200 = .301
a^.301 = 200
.301 loga = log200
a = 44,118,214.29

check: 44118214.29^.301 = 200

unfortunately your log 1000 = .4771 does not even that base.

Check your question.