SOLUTION: What is the largest positive integer n that satisfies n^200

Click here to see ALL problems on Problems-with-consecutive-odd-even-integers

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.

SOLUTION: What is the largest positive integer n that satisfies n^200 < 3^500 ?