Maths
posted by MAths lover Please Helppppp on .
We are given that log102<0.302. How many digits are there in the decimal representation of 5^500?

I am sure you meant:
log_{10} 2 < .302
(you could have just said log 2 ≤ .302 , if the base is omitted it is assumed to be base 10)
let x = 5^500
= (10/2)^500
log x = log (10/2)^500
log x = 500(log (10/2)
= 500(log10  log2)
= 500(1  log2)
= 500  500log2
= 500  151 , = 349 from your given
= 349.48 by using calculator for log2
so your log x > 349
what does that mean?
if logx = 2 , then x = 10^2 100  3 digits
if logx = 3, then x = 10^3  1000  4 digits
...
if logx = 349 , then x = 10^349  350 digits
but log x > 349 , so it must contain 351 digits