perplexus.info :: Just Math : Golden Mean vs Log

Golden Mean vs Log (Posted on 2024-07-11)

Without direct evaluation, or using a calculator, determine which of these is larger:

(1+√5)/2 vs log_e4

See The Solution Submitted by K Sengupta

Rating: 5.0000 (2 votes)

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Possible solution

Comment 1 of 1

since 5>4 we know sqrt(5)>2 and so

(1+sqrt(5))/2 > (1+2)/2 = 3/2

next: we know e>2.7, so

e > 2.56 = 256/100 = 64/25

e^(3/2) > (64/25)^(3/2) = 512/125 > 500/125 = 4

e^(3/2) > 4

ln(4) < 3/2

(1+√5)/2 > 3/2 > log_e4

Posted by Jer on 2024-07-12 11:11:04

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