# If $log\:_{5.2^{x}+1}^{2}$, $log\:_{2^{x-1}+1}^{4}$ and 1 are in HP, then

$(a)\;x\;is\;an\;integer\qquad(b)\;x\;is\;+ve\;real\qquad(c)\;x\;is\;-ve\;real\qquad(d)\;x\;is\;rational\;but\;not\;integer$

Answer : (c) x is -ve real
Explanation : $\large\frac{2}{log\;_{1+2^{1-x}}^{4}}=1+\large\frac{1}{log\;_{5.2^{x}+1}^{2}}$
$(1+2^{1-x})=2\;.(5.2^{x}+1)$
$1+2^{1-x}=10\;.2^{x}+2$
$\large\frac{2}{2^{x}}=10\;.2^{x}+1$
$10\;(2^{x})^{2}+(2^{x})-2=0$
$Taking\;2^{x}=t$
$10t^2+t-2=0$
$10t^2+5t-4t-2=0$
$5t(2t+1)-2(2t+1)=0$
$t=\large\frac{2}{5}\quad\;or\quad\;t=\large\frac{-1}{2}$
$2^{x}=\large\frac{2}{5}\quad\;or\quad\;2^{x}=\large\frac{-1}{2}$
$x=log\;_{2}^{\large\frac{2}{5}}$
$x\;is\;-ve\;real\;.$