Which is Bigger ? No Calculator Allowed

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[@Nikos_Iosifidis]

Very nice solution

[@RealQinnMalloryu4]

5^43or 4^53 5^43^1or 4^53^1 5^1^1 or 4^1^1 5^1 or 4^1 ( x ➖ 5x+1).(x ➖ 4x+1). 5^4311 >4^5311. ❤ amazing video as always.

[@key_board_x]

a = 5^(4311) b = 4^(5311) You know that: 5311 > 4311, so let's calculate the value of x such as: 5^(4311) = 4^(4311 + x) Ln[5^(4311)] = Ln[4^(4311 + x)] 4311.Ln(5) = (4311 + x).Ln(4) 4311.Ln(5) = 4311.Ln(4) + x.Ln(4) x.Ln(4) = 4311.Ln(5) - 4311.Ln(4) x.Ln(4) = 4311.[Ln(5) - Ln(4)] x = 4311.[Ln(5) - Ln(4)] / Ln(4) x = 4311.[ { Ln(5)/Ln(4) } - 1] x ≈ 4311.[ 1.16 - 1] x ≈ 4311 * 0.16 x ≈ 4311 * 0.16 x ≈ 694 5^(4311) = 4^(4311 + x) ← you can write this equality only when x ≈ 694 5^(4311) ≈ 4^(4311 + 694) 5^(4311) ≈ 4^(5005) …but in our case, you have 4^(5311) instead of 4^(5005), i.e. 306 more, so you can write that: 5^(4311) < 4^(5005 + 306) Conclusion: 5^(4311) < 4^(5311)

[@elmer6123]

4^5311/5^4311=(4/5)^4311*4^1000. Take 4000th root. (4/5)^(4311/4000)*4^(1/4)>(4/5)√2>0.8*1.4>1 so 4^5311>5^4311

[@edwinpittomvils2570]

Divide both sides by 4^4311 gives (5/4)^4311 4^1000. Since a=(5/4)^5 is smaller than 3.1