Determine all possible positive real pair(s) (A, B) that satisfy the following system of simultaneous equations:
(3A)log 3 = (8B)log 2, and:
2log A = 3log B * 4log 3
Note: For the purposes of the problem, the base of the logarithm is a positive real number > 1.
Taking logarithms of both sides, the first becomes
log3(log3+logA) = log2(log8+logB)
then solving for B:
logB=(log3logA-log8log2+log3log3) / log2
Doing the same with the second equation:
logAlog2 = logBlog3 + log3log4
logB = (logAlog2 - log3log4) / log3
then plotting exp((ln3lnx-ln8ln2+ln3ln3)/ln2) and exp((lnxln2-ln3ln4)/ln3) on the same graph shows an intersection around x=1/3 (or A=1/3 for the puzzle). Then solving for B, using 1/3 for A, gives 1/8 in either equation, so (1/3, 1/8) is a solution.
From the simultaneous graphs it would appear that (0,0) is also a solution. However, the log of zero is undefined, and so the two equations only approach satisfaction as A and B approach zero.
Posted by Charlie
on 2008-04-04 13:22:33