Question

Solve the equation for $x$
$4^x-3^{x-\frac{1}{2}}=3^{x+\frac{1}{2}}-2^{2x-1}$

Answer 1

$4^x-3^{x-\frac{1}{2}}=3^{x+\frac{1}{2}}-2^{2x-1}$

$4^x+2^{2x-1}=3^{x+\frac{1}{2}}+3^{x-\frac{1}{2}}$

$2^{2x}+2^{2x-1}=3^x(\sqrt{3}+\frac{1}{\sqrt{3}})$

$2^{2x}(1+2^{-1})=3^x\left(\frac{4}{\sqrt{3}}\right)$

$4^x\left(\frac{3}{2}\right)=3^x\left(\frac{4}{\sqrt{3}}\right)$

$\frac{4^x}{3^x}=\left(\frac{4}{\sqrt{3}}\right)\left(\frac{2}{3}\right)$

${\left(\frac{4}{3}\right)}^x=\frac{8}{3\sqrt{3}}$

${\left(\frac{4}{3}\right)}^x={\left(\frac{4}{3}\right)}^{\frac{3}{2}}$

$\Rightarrow x=\frac{3}{2}$