Prove that (log
4 20)
2 + (log
2 3)
2 > 7
P&P solutions only, please.
http://perplexus.info/show.php?pid=7436&op=sol
There are some very good approximations for the common logarithms of 2 and 3. log2=12/40, log3=19/40
Each is too small, but only by a few tenths of a percent.
log(base4)20 = log20/log4 =(1+12/40)/(2*12/40) = (52/24) = 13/6 = 26/12
log(base2)3 = log3/log2 = (19/40)/(12/20) = 19/12
(26/12)^2 + (19/12)^2 = (676+361)/(144) = 1037/144
Since 144*7=1008
1037/144 > 7
1037/1008 is about 1.03 so the errors in the approximations would have to have greatly compounded for this to be wrong.
If I pick up a calculator, the LHS if the inequality is about 7.182 so is greater than 7 by about 2.6%.
Note: the 26/12 approx. is about 0.26% too small and the 19/12 is about 0.10% too big
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Posted by Jer
on 2024-06-27 11:36:14 |
Approx. Solution
