# Maths

posted by .

We are given that log102<0.302. How many digits are there in the decimal representation of 5^500?

• Maths -

I am sure you meant:
log10 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

## Similar Questions

1. ### MATHS

I am less than 500. I am divisible by 11 and 5. I have three digits. The sum of my digits is 14
2. ### discrete math

How many strings of four decimal digits (Note there are 10 possible digits and a string can be of the form 0014 etc., i.e., can start with zeros.) (a) have exactly three digits which are 9s?
3. ### Maths

We are given that log10 2<0.302. How many digits are there in the decimal representation of 5^500?
4. ### Chemistry

When 25.302 is divided by 56.743, the answer should be reported to ____ significant digit(s). The result of the division, 25.302 / 56.743, reported to the correct number of significant digits is _______ When 56.743 is subtracted from …
5. ### maths

1. Log10²x+log10x²=log10² 2-1 2. Log4(log2x)+log2(log4x)=2 3. X^logx+5/3= 10^5+log x 4. Log 1/2(x-1)+ log 1/2(x+1)-log1/√2(7-x)=1
6. ### maths

invaders put a six digit number into the display. you must change each of the six digits to zero in as few turns as possible. on each turn you can use only one number key, the zero key as often as you like and the + key. Example: start …
7. ### Maths

I am less than 500 . I am divisible by 11 and 5. I have three digits. The sum of my digits is 14.
8. ### Math

What am I? I am a decimal fraction with three digits I am bigger than half of 6 I am smaller than double 2 All my digits are odd When you add my digits you get 15
9. ### Math

How many digits are in the decimal representation of 5^40?
10. ### calculus

A positive multiple of 11 is good if it does not contain any even digits in its decimal representation. (a) Find the number of good integers less than 1000. (b) Determine the largest such good integer. (c) Fix b ≥ 2 an even integer. …

More Similar Questions