If a -log 12 18- b -log 24 54 then the value of ab -5 a - b isA- 0B- 4C- 1D- None of these

The correct option is C 1 We have a=log_12 18 =log_218/log_212=1+2 log_23/2+log_23 and b=log_24 54 =log_254/log_224=1+3 log_23/3+log_23. Putting x=log_23, w ...

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Byju's Answer

Standard XI

Mathematics

Fundamental Laws of Logarithms

If a =log 12 ...

Question

If a= $l o g_{12} 18$ , b= $l o g_{24} 54$ then the value of ab+5(a-b) is

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None of these

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Solution

The correct option is C
1

We have

a= $l o g_{12} 18 = \frac{l o g_{2} 18}{l o g_{2} 12} = \frac{1 + 2 l o g_{2} 3}{2 + l o g_{2} 3}$ and

b= $l o g_{24} 54 = \frac{l o g_{2} 54}{l o g_{2} 24} = \frac{1 + 3 l o g_{2} 3}{3 + l o g_{2} 3}$ .

Putting $x = l o g_{2} 3,$ we have $a b + 5 (a - b) = \frac{1 + 2 x}{2 + x} . \frac{1 + 3 x}{3 + x} + (\frac{1 + 2 x}{2 + x} - \frac{1 + 3 x}{3 + x}) = \frac{6 x^{2} + 5 x + 1 + 5 (- x^{2} + 1)}{(x + 2) (x + 3)} = \frac{x^{2} + 5 x + 6}{(x + 2) (x + 3)}$

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If a=log1218, b=log2454 then the value of ab+5(a-b) is

The correct option is C 1 We have a=log1218=log218log212=1+2log232+log23 and b=log2454=log254log224=1+3log233+log23. Putting x=log23, we haveab+5(a−b)=1+2x2+x.1+3x3+x+(1+2x2+x−1+3x3+x)=6x2+5x+1+5(−x2+1)(x+2)(x+3)=x2+5x+6(x+2)(x+3)

If a= $l o g_{12} 18$ , b= $l o g_{24} 54$ then the value of ab+5(a-b) is