To do this question, we need to use the fact that 2 x 3 = 6 to convert the logs.

We obtain log 2 3 from log 6 2, just we need to swap the order of log 6 2 into 1 divided by log 2 6 so that split the log 2 6 into log 2 2 + log 2 3.

Once we obtain log 2 3 (and we are oven log 5 3) we can get log 5 2.

Log 5 2 can be reexpresses using the change of base law as log 3 2 divided by log 3 5 where 3 is the new common base.

How did you know that you need to change to base 3? Why not 2,5 or 6?

The question wants log 5 2. And we are given log 5 3 (the other data is log 2 6). We can’t directly use log 2 6 to solve the problem. Therefore, we need the connector log 2 3 or log 3 2 to meddle with the log 5 2 and log 5 3.

So the number we have to settle with is either the 2 or the 3.

Somehow we need to obtain log 2 3 or log 3 2 from log 2 6 alone as a result. The only way is to split log 2 6 into log 2 2 + log 2 3. The question connects onwards from there.

Fortunately, these questions are unlikely to be seen in the O Levels.

In the end, for the common base, I chose base 3 because the question asks for log 5 2. Neither of the numbers are 3, so the new base I introduce is 3.

Log 5 2 = log 3 2 / log 3 5 by the change of base rule

We have log 5 3 which is the reciprocal of log 3 5 and we have log 2 3 which is the reciprocal of log 3 2.

Alright, thank you very much!