Question

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?

Answer 1

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

Answer 2

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)