If `a=log_(12)18` and `b=log_(24)54` then find the value of `ab+5(a-b)`

Answer

1

Explanation

We have ` a = log_(12) 18= (log_(2)18)/(log_(2)12)=(1+2 log_(2)3)/(2+log_(2)3)`
` and b = log_(24) 54= (log_(2) 54)/(log_(2)24) = (1+3 log_(2)3)/(3+log_(2)3)`
Putting ` x = log_(2) 3`, we have
` ab+5(a-b)=(1+2x)/(2+x)*(1+3x)/(3+x) + 5((1+2x)/(2+x)-(1+3x)/(3+x))`
` = (6x^(2)+5x+1+5(-x^(2)+1))/((x+2)(x+3))`
` = (x^(2)+5x+6)/((x+2)(x+3))=1`