Yahoo Answers: Answers and Comments for Which is bigger? 99^101 or 101^99? [Mathematics]
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From jazzyjem1
enGB
Mon, 22 Jul 2019 07:47:48 +0000
3
Yahoo Answers: Answers and Comments for Which is bigger? 99^101 or 101^99? [Mathematics]
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From rotchm: L := log_10.
99^101 vs 101^99
L(99^101) vs ...
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Mon, 22 Jul 2019 11:56:46 +0000
L := log_10.
99^101 vs 101^99
L(99^101) vs L(101^99)
101L99 vs 99L101
L(99)/99 vs L(101)/101
Consider the function L(x)/x. This is a decreasing function (for x > 3) Hence,
L(99)/99 > L(101)/101. Going back up the steps,
99^101 > 101^99

From Amy: METHOD #1
log(99^101) = 101 log(99) = 201.559...
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Mon, 22 Jul 2019 08:41:03 +0000
METHOD #1
log(99^101) = 101 log(99) = 201.559
log(101^99) = 99 log(101) = 198.42
If log(a) > log(b) then a > b.
METHOD #2
99^101 / 101^99 = (99/101)^99 * 99^2
= 1353...
If a/b > 1 then a > b.
METHOD #3  WITHOUT A CALCULATOR
(combine methods 1 and 2)
log(99^101) / log(101^99)
= (101 log(99)) / (99 log(101))
= (101/99) * (log(99) / log(101))
= (101/99) * (log(99) / log(101))
The log curve is much flatter than linear, so (101/99) > (log(101) / log(99)).
(101/99) * (log(99) / log(101)) > 1

From Mike G: 99^101/101^99 =
1353.12728018
Answer 99^101
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Mon, 22 Jul 2019 08:43:58 +0000
99^101/101^99 =
1353.12728018
Answer 99^101

From sepia: 99^101 > 101^99
99^101
= 3.623720178604972...
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Mon, 22 Jul 2019 08:00:42 +0000
99^101 > 101^99
99^101
= 3.6237201786049720... × 10^201
101^99
=
2.678033494476758508... × 10^198

From Ivan: 99^101 because its being raised by a higher ex...
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Wed, 24 Jul 2019 17:42:34 +0000
99^101 because its being raised by a higher exponent.

From Anonymous: 99¹⁰¹ > 101⁹⁹
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Wed, 24 Jul 2019 00:44:59 +0000
99¹⁰¹ > 101⁹⁹

From Dixon: You can guess the bigger power will win.
The b...
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Mon, 22 Jul 2019 12:38:08 +0000
You can guess the bigger power will win.
The bases are almost the same number (basically 100±1%)
Whereas the powers being different by 2 actually results in factor change of approx 100^2 = 10,000

From Morningfox: 99^101 is bigger
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Mon, 22 Jul 2019 10:14:56 +0000
99^101 is bigger

From Johnathan: You would think that 99^101 < 101^99 just b...
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Mon, 22 Jul 2019 16:28:15 +0000
You would think that 99^101 < 101^99 just because you have a greater base. But the exponents in this case are the bigger telltale. First I'd use simpler powers of the bases. I'll go up to 5 for each of them.
101^1  self explanatory.
101^2  10,201 (5 digits).
101^3  1,030,301 (7 digits).
101^4  104,060,401 (9 digits).
101^5  10,510,100,501 (11 digits).
99^1  self explanatory.
99^2  9,801 (4 digits).
99^3  970,299 (6 digits).
99^4  96,059,601 (8 digits).
99^5  9,509,900,499 (10 digits).
See the patterns? 101^x will have 2x + 1 digits if x is a whole number at least 1, so 101^99 will have 2(99) + 1 = 199 digits. 99^x will have 2x digits with the same restriction for x, so 99^99 will only have 2(99) = 198 digits and won't surpass 101^99. But 99^100 will  it would have 200 digits, and 99^101 would have 202 digits. The verdict:
99^101 > 101^99.

From Vaman: Try this 99^101=99^100 *99. 101^99=101^100/101...
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Mon, 22 Jul 2019 14:18:54 +0000
Try this 99^101=99^100 *99. 101^99=101^100/101. Divide one by the other
(99/101)^100 (101/99). The first term is extremely small. Therefore 101^99 is bigger the number.