Given

$\rm 3^{x + 1} - 3^{x} = 54$

By the law of indices, $\rm a^{m+n} = a^{m} \cdot a^{n}$,

$\rm or, 3^{x} \cdot 3^{1} - 3^{x} = 54$

$\rm or, 3^{x} \left ( 3^{1} - 1 \right ) = 54$

$\rm or, 3^{x} \left ( 3 - 1 \right ) = 54$

$\rm or, 3^{x} \cdot 2 = 54$

Dividing both sides of the equation by 2, we get,

$\rm or, 3^{x} \cdot \frac{2}{2} = \frac{54}{2}$

$\rm or, 3^{x} = 27$

$\rm or, 3^{x} = 3^{3}$

The bases of the terms on both sides of the equation are the same, so we equate their powers.

$\rm \therefore x = 3$

Hence, the required value of x is 3.