Since I don't know the 'yana girl story', I can only guess. It could be about Yana's dream pursuit. For example, she might be a young girl with big dreams in a small town, and the story could be about how she overcomes obstacles like lack of resources, opposition from some people in the town, and her own self - doubts to finally achieve her dreams. It could also be a love story where Yana falls in love with someone unexpected and how they deal with various situations together.
Again, without knowing the actual 'yana girl story', I can't say for sure. But generally, main elements in a story could be the character of Yana herself, the setting where the story takes place, and the plot events that happen to her.
The fanfic might revolve around a day in their lives. Ice Bear could be showing Yana his secret hiding places in the city, like a rooftop garden he found. They could share stories and learn more about each other's pasts, with Yana being amazed by Ice Bear's survival skills.
They could both develop a deeper bond of friendship. Through shared experiences like helping out at a local animal shelter or solving a mystery in their neighborhood, they would trust each other more. This would lead to them being more comfortable around each other and being able to communicate without words in some situations.
I'm not sure who the girl is in this 'girl in the log story'. It could be a young girl from a small village. She might be an adventurous type who is always exploring the woods around her village. And the log could be something she stumbles upon during one of her adventures, and from there, the story unfolds about her experiences with this log.
There's no way to tell for sure without more details. She could be an ordinary girl who becomes extraordinary through the events in the book, or she could be a princess, a peasant, or a girl with special powers.
Since there's no more information, I can't say for sure who the girl is. She could be anyone - a fictional character created for the story or a real person.