Don't worry,no question will be asked to find log of a random number.
In most cases the answer it self will be in log.
In many cases,the value of log that you need will be provided in the question itself.
But remember that JEE is a competitive exam and you should be well prepared.You cannot take any chances.
So the best thing you can do is,try to by heart log of 2,3 and 5.
From which you can round off many log values.
Some times the log value given with the question will not be the actual value.you need to do basic log operation with 2,3 etc to find the actual value.
So my advice is learn the log values of 2,3 and 5 which will be very handy during exams.
Remember that JEE is an all India exam,your weakness in certain area is enough for your failure.