Question

Solution of the equation $1+3^{\frac{x}{2}}=2^x$ is

Answer 1

We have
$1+3^{\frac{x}{2}}=2^x$
Dividing both sides by $2^x$, we obtain
$\Rightarrow \frac{1}{2^x}+\frac{3^{\frac{x}{2}}}{2^x}=1(\because 2^x>0)$
$\Rightarrow {\left(\frac{1}{2}\right)}^x+{\left(\frac{3}{4}\right)}^{\frac{x}{2}}=1\Rightarrow {\left(cos\frac{\pi }{3}\right)}^x+{\left(sin\frac{\pi }{3}\right)}^x=1$
which is possible the only when $x=2$
Hence, $x=2$ is only solution of the original equation .