How many different numbers between 1/1000 and 1000 can be written either as a power of 2 or as a power of 3, where the exponent is an integer?

FUN Q!

look for 1/1000 < x < 1000 where x=2^n, n must be an integer

taking log on the inequalities

log1/1000 < logx < log1000

-3 < logx < 3

take log on x=2^n

logx=log (2^n) = nlog2=0.301n

substituting

-3 < 0.301n < 3

-9.9658 < n < 9.9658

n must be an integer: - 9, - 8, ... 0, 1, ... 9

ans is 19

u can repeat the same with log (3)

following same reasoning as the above ans, log (1/1000) < log (3^n) < log (1000)

-3 < n*log3 < 3

-3 < 0.48n < 3

-6.3 < n < 6.3

n can be - 6,-5, ... 5,6

total of 13 possible numbers for 3^n