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If a=(log)(12)18 , b=(log)(24)54 , then ...

If `a=(log)_(12)18 , b=(log)_(24)54 ,` then find the value of `a b+5(a-b)dot`

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The correct Answer is:
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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`
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