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log base 24 54 = a and log base 12 18 = b Then the value of [ab-5(a-b)].

log base 24 54 = a  and  log base 12 18 = b
Then the value of [ab-5(a-b)].

Grade:9

1 Answers

Arun Kumar IIT Delhi
askIITians Faculty 256 Points
7 years ago
Hello Student,

\\a=log_{24}54={ln54 \over ln24}={ln3+ln18 \over ln2+ln12} \\a={3ln3+ln2 \over 3ln2+ln3}={3x+y \over 3y+x} \\b=log_{12}18={ln18 \over ln12} \\={2ln3+ln2 \over 2ln2+ln3}={2x+y \over 2y+x} \\=>ab-5(a-b) \\=>{3x+y \over 3y+x}*{2x+y \over 2y+x}-5({3x+y \over 3y+x}-{2x+y \over 2y+x}) \\=>{y^2+5xy+6x^2-5((3x^2+2y^2+7xy)-(3y^2+2x^2+7xy)) \over (2y+x)*(3y+x)} \\=>{y^2+5xy+6x^2 - 5(x^2-y^2) \over (2y+x)*(3y+x)} \\=>{6y^2+5xy+x^2 \over 6y^2+x^2+5xy} \\=>1
Thanks & Regards
Arun Kumar
Btech, IIT Delhi
Askiitians Faculty

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