Step 1: Understand the base rule.
Each letter is replaced by its position in the alphabet (A=1, ..., Z=26), and then every vowel gets an extra $+2$ before being written.
Step 2: Write the plain positions of LOGIC.
L=12, O=15, G=7, I=9, C=3.
Step 3: Mark the vowels.
In LOGIC the vowels are O and I; L, G, C are consonants and stay unchanged.
Step 4: Shift the vowel O.
O is $15$, so $15 + 2 = 17$.
Step 5: Shift the vowel I.
I is $9$, so $9 + 2 = 11$.
Step 6: Assemble the code.
Putting the values in order: $12, 17, 7, 11, 3$, which is $12$-$17$-$7$-$11$-$3$.
\[ \boxed{12\text{-}17\text{-}7\text{-}11\text{-}3} \]