Question

If ${\log }_{\left(5.2^x+1\right)}2;{\log }_{\left(2^{1-x}+1\right)}4$ and 1 are in Harmonical Progression then

• A. $x$ is a positive real
• B. $x$ is a negative real
• C. $x$ is rational which is not integral
• D. $x$ is an integer

Answer 1

Answer: (B)

$a,b,c$ are in H.P. $\Rightarrow b=\frac{2ac}{a+c}\Rightarrow \frac{\log 4}{\log \left(2^{1-x}+1\right)}=\frac{2\cdot \frac{\log 2}{\log \left(5\cdot 2^x+1\right)}\cdot 1}{\frac{\log 2}{\log \left(5\cdot 2^x+1\right)}+1}$
$\frac{2\log 2}{\log \left(2^{1-x}+1\right)}=\frac{2\log 2}{\log \left(5\cdot 2^x+1\right)[\log 2+\log \left(5\cdot 2^x+1\right)}$
$.t+2=2/t+1\Rightarrow 10t^2+2t=2+t\left(2^x=t\right)$
$10t^2+t-2=0$
$10t^2+5t-4t-2=0$
$5t(2t-1)-2(2t+1)=0\Rightarrow t=2/5,-1/2$
$x\log 2=\log 2/5\Rightarrow 2^x=2/5$
$x{\log }_{2}2=1-{\log }_{2}5$
$x=1-{\log }_{2}5.$