Aspire Faculty ID #16176 · Topic: NIMCET 2011 · Just now
NIMCET 2011

$a,b,c$ are positive and $c>a$ and in H.P. Compute $\log(a+c)+\log(a-2b+c)$.

Solution

$a,b,c$ in H.P. ⇒ 

$\dfrac{1}{b} = \dfrac{1}{2}\left(\dfrac{1}{a}+\dfrac{1}{c}\right)$ 

⇒ $2bc = ac + ab$ 

Simplify gives identity: 

$(a+c)(a-2b+c) = (c-b)^2$ 

 Take log: 
$\log(a+c) + \log(a-2b+c) $
$= \log\big((c-b)^2\big)$
$=2\log(c-b)$
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