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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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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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CENGAGE-LOGARITHM AND ITS PROPERTIES-Exercise (Numerical)
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  2. The value of (log(10)2)^3+log(10)8log(10)5+(log(10)5)^3 is ..............

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  3. If (log)4A=(log)6B=(log)9(A+B),t h e n [4(B/A] (where [ ] represents t...

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  4. Integral value of x which satisfies the equation =log6 54+(log)x 16=(l...

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  5. If a = log(245) 175 and b = log(1715) 875, then the value of (1-ab)/...

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  6. The difference of roots of the equation ((log)(27)x^3)^2=(log)(27)x^6 ...

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  7. Sum of all integral values of x satisfying the inequality (3^((5/2)log...

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  8. The least integer greater than (log)2 15 (log)(1/6)2(log)3 1/6 is .......

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  9. The reciprocal of 2/((log)4(2000)^6)+3/((log)5(2000)^6) is ..............

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  10. Sum of integers satisfying sqrt((log)2x-1)-1/2(log)2(x^3)+2>0 is......

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  11. Number of integers satisfying the inequality (log)(1/2)|x-3|>-1 is.......

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  12. Number of integers lt=10 satisfying the inequality 2(log)(1/2)(x-1)lt=...

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  13. The value of (sqrt(3+2sqrt2)+sqrt(3-2sqrt2))^(2^(9)) is .

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  14. The value of 5^((log)(1/5)(1/2))+(log)(sqrt(2))4/(sqrt(7)+sqrt(3))+(lo...

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  15. The value of N=((log)5 250)/((log)(50)5)-((log)5 10)/((log)(1250)5) is...

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  16. if x+log10 (1+2^x)=xlog10 5+log10 6 then x

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  17. The x , y , z are positive real numbers such that (log)(2x)z=3,(log)(5...

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

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