Log magic squares

[Guest]

given the magic square below your task is to express xyz in terms of abc

loga logb logx
logp logy logc
logz logq logr

1. for each row column and diagonal create an equation representing the sum
2. use the log laws to simplify equations
3. convert each equation to exponential form
4. combine the equations somehow (use substitution or elimination) to create an equation that expressed the product xyz inn terms of abc

[Guest]

It is easy to check that the following all hold
[tex]pyc = abx[/tex]
[tex]xyz = abx[/tex]
[tex]xyz = ayr[/tex]
[tex]xyz = ayr[/tex]
[tex]xcr = apz[/tex]
[tex]xcr = abx[/tex]

Multiply the equations to get
[tex]c^3 p r^2 x^5 y^4 z^3 = a^6 b^3 p r^2 x^3 y^2 z[/tex]
which simplifies to
[tex]x^2 y^2 z^2 = a^6 b^3 c^{-3}[/tex]
giving us the expression
[tex]xyz = a^3 b^{1.5} c^{-1.5}[/tex]

This is the best we can do [tex]xyz[/tex] cannot be expressed purely as an expression of [tex]abc[/tex] (to see this simply consider the magic squares that are made from the numbers 1,...,9).

R. Baber.

[Guest]

Thank you Mr. Baber

[namal]

Thanks Baber
_______________________
namal