\hspace{-16}$Find value of $\mathbf{x}$ in $\mathbf{2^x+2^{\left[x\right]}+2^{\left\{x\right\}}=3}$\\\\ Where $\mathbf{\left[.\right]=}$ Greatest Integer Function and \\\\ $\mathbf{\left\{.\right\}=}$ fractional part function.
1yes x = 0is the only possible solution.
obviously x will be positive. (plug in to check)
take cases,
1: 0≤x<1
...here [x] = 0 and {x} = x
this gives us x = 0.
2: <1x<2
...here [x] = 1 and {x} = x-1
this gives us a negative value to 2x hence not possible..so no need to take any further cases.
1708also
\hspace{-16}$When $\mathbf{x\in [-1,0)}$\\\\ Then $\mathbf{x=\ln_{2}\left(\frac{5}{6}\right)}$