Question

If xy^(a-1)=l, xy^(b-1)=m, xy^(c-1)=n, then prove that a log m/n+b log n/l+c log l/m=0​

Answer 1

Answer:

There is a logarithmic property that is very useful to know: logb(a)=1loga(b). This can be proven using the change of base formula, but it means that 1logxy(xyz)=logxyz(xy). Likewise, 1logyz(xyz)=logxyz(yz) and 1logzx(xyz)=logxyz(zx).

This means that the original expression can be written as logxyz(xy)+logxyz(yz)+logxyz(zx).

Adding the logarithms gives us logxyz(xy×yz×xz)

=logxyz(x2y2z2)

=logxyz((xyz)2)

=2.