To solve for the missing exponent in the equation
\[ \frac{e^{\text{unknown exponent}}}{e^{65}} = e^{38}, \]
we can start by letting \( x \) be the unknown exponent. Thus, the equation can be rewritten as:
\[ \frac{e^x}{e^{65}} = e^{38}. \]
Using the properties of exponents, we know that:
\[ \frac{e^x}{e^{65}} = e^{x - 65}. \]
So now we have:
\[ e^{x - 65} = e^{38}. \]
Since the bases \( e \) are the same, we can set the exponents equal to each other:
\[ x - 65 = 38. \]
To find \( x \), we can add 65 to both sides:
\[ x = 38 + 65 = 103. \]
Thus, the missing exponent is
\[ \boxed{103}. \]