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Fundamental Laws of Logarithms

If a=log12 18...

Question

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

A

\N

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B

4

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C

1

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D

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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Similar questions

a = {log}_{12} 18

b = {log}_{24} 54

, then find the value of

a b + 5 (a - b)

a = {log}_{12} 18

b = {log}_{24} 54,

then the value of

a b + 5 (a - b)

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

l o g_{12} 18 = a

l o g_{24} 54 = b

a b + 5 (a - b)

l o g_{12} 18 = α

l o g_{24} 54 = β

α β + 5 (α - β) = 1

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