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.

Charlie's idea to take the log of both sides of each equation is immensely helpful. I'm rewriting it here with the following substitutions:

Let a = logA, b=logB, x=log2, y=log3 (this makes the equations easier to read. Then,

y(y+a) = x(3x+b) [since log8 = 3log2]

xa = yb+2xy [since log4 = 2log2]

using the second equation,

a=yb/x + 2y

substituting into the first equation gives

y(yb/x+3y) = x(3x+b)

y^2b/x + 3y^2 = 3x^2 + bx

(y^2 - x^2)b/x = 3(x^2-y^2)

b/x = -3

b=-3x

and then substituting back into the expression for a:

a= -3xy/x + 2y = -3y+2y

a=-y

rewriting back in terms of logs,

logB = -3log2 = log(1/8) so B=1/8

logA = -log3 = log(1/3) so A=1/3

this agrees with the graphical solution, lending credence to both.

Posted by Paul on 2008-04-15 04:25:57