Calculate directly (without using a calculator)?

The log of a product is the sum of the logs of the factors.

For example, if we have to find the log of ab (a times b), we can simply find the log of a and add it to the log of b

log(ab) = log(a) + log(b)

(The operator that allows us to change of operation into another is called a "transformation"; the logarithm allows us to "transform" a multiplication into an addition).

I imagine log3 means "logarithm in base 3"

log3(a) + log3(b) = log3(ab)

log3(9/8) + log3(72) = log3[(9/8)(72)] = log3(9*9) = log3(81) = log3(3*3*3*3)

= log3(3) + log3(3) + log3(3) + log3(3)

And, of course, the log of the base = 1.

for example

ln(e) = 1 (ln = natural logs in base e)

log10(10) = 1

log7(7) = 1

and so on.

if you are adding logs with the same base number you simply multiply them (it's one of the rules of logs)

So..

log3(9/8) + log3(72) = log3 (9*72/8)

= log3(81)

This simplifies as log3(81) is log3(3)^4...

Rule = Loga(a) = 1

So ... log3(3)^4 = 1^4 which is 1!

Answer: 1

Look up John Napier, who in about 1600 calculated accurate values of logs in Scotland. The Open University has a good article. Also look up "calculation of logarithms".