Let's denote the original number Uche thinks of as \( x \). According to the problem, Uche adds 7 to \( x \) and then doubles the result. The equation can be set up as follows:
- Add 7 to \( x \): \( x + 7 \)
- Double the result: \( 2(x + 7) \)
We are given that this final result equals 34:
\[ 2(x + 7) = 34 \]
Now, we can solve for \( x \):
- Divide both sides of the equation by 2:
\[ x + 7 = 17 \]
- Subtract 7 from both sides:
\[ x = 17 - 7 \]
\[ x = 10 \]
So, the original number Uche thought of is \( \boxed{10} \).