Bài giải:

2.2².2³....2^x =32768

$\Rightarrow $ 2.2².2³....2^x = 2¹⁵

$\Rightarrow $2^1+2+3+...+x = 2¹⁵

Ta có: 1 + 2 + 3 + ... + x = (1 + x).$\frac{x}{2}$

$\Rightarrow $ (1 + x).$\frac{x}{2}$ = 15

$\Rightarrow $ (1 + x).x = 30

$\Rightarrow $ x + x² – 30 = 0

$\Rightarrow $ x² + x – 30 = 0

$\Rightarrow $ (x – 5)(x + 6) = 0

$\Rightarrow $$\left\{ \begin{align}& x-5=0 \\ & x+6=0 \\ \end{align} \right.$

$\Rightarrow $$\left\{ \begin{align}& x=5 \\ & x=-6 \\ \end{align} \right.$

Mà: x $\ge $ 1

$\Rightarrow $ x = 5

Vậy: x= 5