Question

log18base12 = a and log54base24 = b, then prove that ab + 5(a-b) = 1

Answer 1

lets convert all to same base, say m
=> log 18 / log 12 = a= (2log3 +log2)/(2log2 +log3)
=> log 54/ log 24 =b = (3log3 +log2)/(log3 + 3log2)
i dont know,,maybe try now..

Answer 2

lets assume log 2 =x and log 3=y
so
(2y + x)/(2x +y) =a
(3y +x)/(y+3x) =b
a-b = (2y^2 + 7xy +3x^2) - (7xy + 3y^2 + 2x^2) / (something)
= 2(y^2 -x^2) -3(y^2 -x^2) / (5xy +y^2 + 6x^2)
= (x^2 -y^2)/(5xy +y^2 + 6x^2)
tired now :P
you may try further..