Determine all possible positive real pair(s) (A, B) that satisfy the following system of simultaneous equations:

(3A)^{log 3} = (8B)^{log 2}, and:

2^{log A} = 3^{log B} * 4^{log 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