Without direct evaluation, determine which of these expressions has a greater numerical value:
2√10 OR 32
Here's one which involves computing ln(2) and ln(3):
2^(sqrt(10)) vs 3^2
(10^.5) ln2 vs 2 ln3
ln(10^.5) + lnln2 vs ln2 + lnln3
.5(ln2 + ln5) + lnln2 vs ln2 + lnln3
(.5)(ln5) + lnln2 vs (.5)(ln2) + lnln3
ln5 + 2*lnln2 vs ln2 + 2*lnln3
ln5 + ln((ln2)^2) vs ln2 + ln((ln3)^2)
ln(5*(ln2)^2) vs ln(2*(ln3)^2)
5*(ln2)^2 vs 2*(ln3)^2
5/2 vs (ln3/ln2)^2
2.5 vs (1.09861... / 0.69314...)^2
2.5 vs (1.58496...)^2
2.5 vs 2.512106...
So,
2^(sqrt(10)) < 3^2
Direct computation to confirm:
8.95241961947087 < 9
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Posted by Larry
on 2023-01-18 13:46:21 |
here's one
