It is convenient that this problem has a whole number solution. In order to solve `2^x=64` without a calcultor you start listing the multiples of 2 and hope that 64 is one of them.
You don't indicate what level of math class you are in. If this is a college algebra class, you will need the methods referenced by etotheeyepi and elekzy -- namely logarithms and exponentials. If this is an algebra class, perhaps dealing with basic exponentials or sequences and series, you will be able to use a guess and check strategy without a calculator.
But solving `2^x=18` without a calculator would be a daunting task indeed.
**Note that both of the above answers assumed you knew the answer. They both used `64=2^6` before solving for x. The correct method using logarithms is:
`x=(ln64)/(ln2)=6` , but this requires a calculator or log tables.