Let the three equal logarithms all be equal to a number we'll call n.
Translating the equations into exponential form, we get:
x=9^n
y=12^n
x+y=16^n
y/x = 12^n/9^n = (4/3)^n
(x+y)/y = 16^n/12^n = (4/3)^n
So (x+y)/y = y/x.
Then y^2  xy  x^2 = 0
y = (x +/ sqrt(5*x^2))/2
y/x = (1 +/ sqrt(5))/2
Since y > x, y/x = (1+sqrt(5))/2, the golden ratio.
The numeric way would be via a computer program:
DEFDBL AZ
n = 1.5
DO
pn = n
n = LOG(9 ^ n + 12 ^ n) / LOG(16)
LOOP UNTIL pn = n
PRINT n, 12 ^ n / 9 ^ n, 9 ^ n / 12 ^ n, 9 ^ n, 12 ^ n, 16 ^ n
The second number printed in the results below is the desired answer, the first being n, and then the second line has x, y and x+y:
1.672720934462332 1.618033988749894 .6180339887498949
39.46221170149797 63.85119980426752 103.3134115057655

Posted by Charlie
on 20041208 15:29:24 