If log 18 to the base 12 = a and log 54 to the base 24 = b , prove that ab+ 5 (a-b) = 1 .

See, log18base12 =a can be written as 12^a=18 2^2a×3^a=2×3²2^2a-1=3^2-a2^2a-1/2-a=3........1And log54base24 =b can be written as 24^b=542^3b×3^b=2×3³2^3b-1=3^3-b2^3b-1/3-b=3........2Equate 1&22^2a-1/2-a= 2^3b-1/3-b2a-1/2-a=3b-1/3-b(2a-1)(3-b)=(3b-1)(2-a)6a-2ab-3+b= 6b-2-3ab+a6a-a-2ab+3ab+b-6ab=-2+35a-5b+ab=15(a-b)+ab=1