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Question Number 193230 by MATHEMATICSAM last updated on 07/Jun/23

$$\mathbf{Choose}\mathbf{the}\mathbf{correct}\mathbf{option}:$$$$\mathrm{If}a,b\mathrm{and}c\mathrm{are}\mathrm{consecutive}\mathrm{positive}$$$$\mathrm{integers}\mathrm{and}\log (1+ac)=2k\mathrm{then}\mathrm{the}$$$$\mathrm{value}\mathrm{of}k\mathrm{is}:$$$$\mathrm{a})\log a$$$$\mathrm{b})\log b$$$$\mathrm{c})2$$$$\mathrm{d})1$$$$\mathbf{Give}\mathbf{the}\mathbf{explaination}\mathbf{also}.$$

Answered by Rajpurohith last updated on 08/Jun/23

$$leta=b-1;c=b+1(sincea,bandcareconsecutive.)$$$$so1+ac=1+(b-1)(b+1)=1+b^2-1=b^2$$$$\Rightarrow log(1+ac)=log\left(b^2\right)=2log\left(b\right)=2k$$$$hencek=log\left(b\right).$$

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