SOLUTION: What is the largest positive integer n that satisfies n^200 < 3^500 ?
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Question 360088: What is the largest positive integer n that satisfies n^200 < 3^500 ?
Found 2 solutions by Jk22, Theo:
Answer by Jk22(389) (Show Source): You can put this solution on YOUR website!
3^500 = 3^(200*2.5) = (3^2.5)^200
3^2.5 = 3^(5/2) = Sqrt(3^5) = Sqrt(3*81) = Sqrt(243)
Square root 243 :
2|43 | 15.x
--------
1|00 | 1
---------
143 | 25x5=125, 26x6=120+36>143
125 |
----------
18
...
hence the number is n=15
other way : 200*log(n) < 500 * log(3)
calculator gives : exp(500*log(3)/200) = 15.58
since n is integer, n = 15.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
n^200 < 3^500
log (n^200) < log(3^500)
200 * log(n) < 500 * log(3)
log (n) < 500*log(3) / 200
log(n) < (500 * .477121255) / 200
log(n) < 1.192803137
n < 15.58845727
When n = 15.58845727:
15.58845727^200 = 3^500
log(15.58845727^200) = log(3^500)
200 * log(15.58845727) = 500 * log(3)
200 * 1.192803137 = 500 * .477121255
238.5606274 = 238.56062874
This is good.
Now take n < 15.58845727.
Use 15.58845720.
Then:
15.58845720^200 < 3^500
200*log(15.58845720) < 500*log(3)
238.560627 < 238.56062874
Looks good to me.
The numbers were too big for the calculator without using logs, so the only way to easily confirm was to take the log and then compare the results.
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