If log 18 to the base 12 =alpha and log 54 to the base 24 = beta , Prove that alpha x beta +5(alpha-

Martha

If log 18 to the base 12 =alpha and log 54 to the base 24 = beta , Prove that alpha x beta +5(alpha-beta)=1

Deepak Aggarwal

α = log₂18 / log₂12 = 1 + 2log₂3 / 2 + log₂3

β = log₂54 / log₂24 = 1 + 3log₂3 / 3 + log₂3

You can now just assume that log₂3 = t,

So, αβ + 5(α-β) = (1 + 2t / 2 + t) + (1 + 3t) / (3 + t) + 5 [ (1 + 2t / 2 + t) - (1 + 3t / 3 + t) = 1

Hence, Proved.