#### Answer

$x = 5$

#### Work Step by Step

To solve the given exponential equation, make the two sides have the same base. Then, equate the exponents and solve for the unknown variable.
Since $27=3^3$, the given equation is equivalent to:
$3^{2x-7} = 3^3$
Use the rule $a^m = a^n \longrightarrow m=n$ to obtain:
$2x-7 = 3$
Add $7$ to both sides to obtain:
$2x = 10$
Divide $2$ to both sides to obtain:
$x = 5$