Concept:

​Logarithm properties:

Calculation:

a = \(\rm \frac {\log18}{\log12} = {\log (3^2×2)\over \log (2^2×3)}\)

⇒ a = \(\rm {2\log 3+\log2\over 2\log 2+\log3}\) 

b = \(\rm \frac {\log54}{\log24} = {\log (3^3×2)\over \log (2^3×3)}\)

⇒ b = \(\rm {3\log 3+\log2\over 3\log 2+\log3}\)

Let log 2 = x and log 3 = y

S = ab + 5(a - b)

⇒ S = \(\rm {2\log 3+\log2\over 2\log 2+\log3}\times\rm {3\log 3+\log2\over 3\log 2+\log3}\) + 5\(\rm \left({2\log 3+\log2\over 2\log 2+\log3} - \rm {3\log 3+\log2\over 3\log 2+\log3}\right)\)

⇒ S = \(\rm {2y+x\over 2x+y}\times\rm {3y+x\over 3x+y}\) + 5 \(\rm\left( {2y+x\over 2x+y}-\rm {3y+x\over 3x+y}\right)\)

⇒ S = \(\rm {6y^2 +5xy+x^2\over 6x^2+5xy+y^2}\) + 5 \(\rm\left( {3x^2+7xy+2y^2-(2x^2 +7xy+3y^2)\over6x^2+5xy+y^2}\right)\)

⇒ S = \(\rm {6y^2 +5xy+x^2\over 6x^2+5xy+y^2}\) + 5 \(\rm\left( {x^2-y^2\over6x^2+5xy+y^2}\right)\)

⇒ S = \(\rm {6y^2 +5xy+x^2\over 6x^2+5xy+y^2}\) + \(\rm {5x^2-5y^2\over6x^2+5xy+y^2}\)

⇒ S = \(\rm {6y^2 +5xy+x^2+5x^2-5y^2\over 6x^2+5xy+y^2}\)

⇒ S = \(\rm {6x^2 +5xy+y^2\over 6x^2+5xy+y^2}\)

⇒ S = 1