Ternary is a base 3 system. We can use a type of long division to get the answer.
We are going to divide 1999 by 3 and just note the remainder:
1999/3=666 rem 1. So 1 is the last digit in the ternary number.
Now divide 666 by 3=222 remainder 0. This goes in front of the 1.
222/3=74 remainder 0. That's another 0 to place in front of 01.
74/3=24 remainder 2. The ternary now ends 2001.
24/3=8 remainder 0. Ternary ends 02001.
8/3=2 remainder 2. Ternary ends 202001.
The last quotient is smaller than 3 so we just make it the first digit of the answer:
2202001. To check this we can use the powers of 3: 1, 310=103, 910=1003, 2710=10003, 8110=100003, 24310=1000003, 72910=10000003.
20000003 is 2×729=145810; 200000=2×243=48610; 20003=2×27=543; 1 is the same in both bases.
So 22020013=1458+486+54+1=199910.