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The terms of a geometric progression are real and positive. If the \(p\)-th term of the progression is \(q\) and the \(q\)-th term is \(p\), then the logarithm of the first term is

Question and options

The terms of a geometric progression are real and positive. If the pp-th term of the progression is qq and the qq-th term is pp, then the logarithm of the first term is
  1. (1−q)log⁡(p)−(1−p)log⁡(q)p−q\dfrac{(1-q)\log(p)-(1-p)\log(q)}{p-q}
  2. (1−q)log⁡(q)−(1−p)log⁡(p)p−q\dfrac{(1-q)\log(q)-(1-p)\log(p)}{p-q}
  3. (1−q)log⁡(p)+(1−p)log⁡(q)p−q\dfrac{(1-q)\log(p)+(1-p)\log(q)}{p-q}
  4. (1−q)log⁡(q)+(1−p)log⁡(p)p−q\dfrac{(1-q)\log(q)+(1-p)\log(p)}{p-q}

Answer and explanation

Correct answer: B - \(\dfrac{(1-q)\log(q)-(1-p)\log(p)}{p-q}\)
Let first term a, ratio r: log a+(p-1)log r=log q and log a+(q-1)log r=log p. Subtracting gives log r=(log q-log p)/(p-q); substituting, log a=[(1-q)log q+(p-1)log p]/(p-q)=[(1-q)log q-(1-p)log p]/(p-q).

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