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Choose-the-correct-option-If-a-b-and-c-are-consecutive-positive-integers-and-log-1-ac-2k-then-the-value-of-k-is-a-log-a-b-log-b-c-2-d-1-Give-the-explaination-also-




Question Number 193230 by MATHEMATICSAM last updated on 07/Jun/23
Choose the correct option:  If a, b and c are consecutive positive  integers and log(1 + ac) = 2k then the  value of k is:  a) log a  b) log b  c) 2  d) 1  Give the explaination also.
$$\boldsymbol{\mathrm{Choose}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{correct}}\:\boldsymbol{\mathrm{option}}: \\ $$$$\mathrm{If}\:{a},\:{b}\:\mathrm{and}\:{c}\:\mathrm{are}\:\mathrm{consecutive}\:\mathrm{positive} \\ $$$$\mathrm{integers}\:\mathrm{and}\:\mathrm{log}\left(\mathrm{1}\:+\:{ac}\right)\:=\:\mathrm{2}{k}\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:{k}\:\mathrm{is}: \\ $$$$\left.\mathrm{a}\right)\:\mathrm{log}\:{a} \\ $$$$\left.\mathrm{b}\right)\:\mathrm{log}\:{b} \\ $$$$\left.\mathrm{c}\right)\:\mathrm{2} \\ $$$$\left.\mathrm{d}\right)\:\mathrm{1} \\ $$$$\boldsymbol{\mathrm{Give}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{explaination}}\:\boldsymbol{\mathrm{also}}. \\ $$
Answered by Rajpurohith last updated on 08/Jun/23
let a=b−1 ; c=b+1 (since a ,b and c are consecutive.)  so 1+ac=1+(b−1)(b+1)=1+b^2 −1=b^2   ⇒log(1+ac)=log(b^2 )=2log(b)=2k  hence k=log(b).
$${let}\:{a}={b}−\mathrm{1}\:;\:{c}={b}+\mathrm{1}\:\left({since}\:{a}\:,{b}\:{and}\:{c}\:{are}\:{consecutive}.\right) \\ $$$${so}\:\mathrm{1}+{ac}=\mathrm{1}+\left({b}−\mathrm{1}\right)\left({b}+\mathrm{1}\right)=\mathrm{1}+{b}^{\mathrm{2}} −\mathrm{1}={b}^{\mathrm{2}} \\ $$$$\Rightarrow{log}\left(\mathrm{1}+{ac}\right)={log}\left({b}^{\mathrm{2}} \right)=\mathrm{2}{log}\left({b}\right)=\mathrm{2}{k} \\ $$$${hence}\:{k}={log}\left({b}\right). \\ $$

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