Just wanted to clarify. The cost of an airship trip is based on your highest level class. From level 1 to 25, the trip on an airship will cost 2,500 gil. The price goes up 100 gil for every level from 26 to 50, at which point it costs 5,000 gil.
Because Mirror's highest class is level 16, the airship trip will cost Mirror 2,500 gil.
Also, Mirror, if you want to buy the logs, I'd suggest checking the Market Wards in Uldah. Uldah is the city located between Limsa Lominsa and Gridania, and most players have migrated there. So the Market Wards in Uldah tend to be more full and active. So there's a much better chance that the logs are being sold in the Uldah Market Wards than in Limsa.
If you're going to make the run, Aleczan has given you some good tips. If it helps, like Aleczan said, you're going to get on the ferry out of Limsa Lominsa. Open your city map, go to Lower Decks, and over on the far West side of the map is a picture of an anchor. Go there, exit through the door, and when the ferry arrives, you can get on it.
You'll then arrive in Thanalan. That's the region where Uldah is located. Here's a map of Thanalan. That's basically the same map you'll see in game. You'll arrive where the anchor is over on the far left side of the map. See that little white path running from the left to right of the map? You're going to want to follow that road all the way across.
In the middle of the map, the road makes a T. On the linked map, there's a red circle, and the T is in the middle of the circle. As long as you're running past it, you'll probably want to stop in Uldah and touch the aetheryte there, so you can teleport back to Uldah whenever you want. So when you come to the T, run to the South until you arrive at the front gates of Uldah.
When you get back on your way to Gridania, you're going to want to follow the road all the way to the East where that upside down yellow triangle is located on the linked map. That will zone you into The Black Shroud. That's the region where Gridania is located.
You're going to enter the Black Shroud all the way at the bottom of this map. (You can scroll that map down until you can see the yellow triangle where you'll enter. It's at the far Southern tip.) You're going to follow the road all the way up until you get to the yellow triangle which, when you hover over it, says "Gridania." It's at the top of the red circle in the linked map.
Good luck!