How can I brighten up a dark log cabin?

I would use light and bright colors. If you are painting the walls, stick to really light colors, perhaps semigloss so any light can be reflected off the walls to make things look brighter. Use colorful, bright accents throughout. For windows, if you are going with drapes, have sheers underneath so that you can open the drapes and let the sunlight in, but still have privacy with the shears if you need it.

I would paint the walls a light color and put some art or wall hangings up so that the walls don't look too empty or white. Coordinate your wall art with other things in your home.

Okay, but could you use more sheers (or drapes?) and hang them on all the walls? Also, you could try adding mirrors or a tapestry.

https://www.apartmenttherapy.com/drapes-across-a-wall-99898

https://freshome.com/wall-decorating-ideas/

Lighting is important. Can you paint your cabinets? Adding even throw pillows in a light color can help. You can find pillow covers that are cheaper than new pillows. Plants will also add color.

You can increase lighting by using LED lightbulbs with a Kelvin rating of 3000/a lumen rating exceeding 1000, or 4 or more bulbs exceeding 900 lumens, in a room measuring 12 x 20, with 8 foot ceiling. Install LED floods with dimmer switches into ceiling fixtures, if ceilings are very high.

Good lighting and lots of mirrors! If you have painted walls, paint them a light color. Use light color furniture and accessories.

I can't paint as I'm renting and really love the log cabin feel.

Because you're renting, you may not be permitted to hang mirrors. If that's the case, purchase a large floor mirror to lean against a wall, where it can bounce light. Mirror should be placed perpendicular to the window, rather than across from it.

Some landlords are obsessive about not putting nail holes into walls. If you can find a means of suspending hooks from the wooden log faces (similar to Victorian picture hangers that were suspended near the ceiling's mouldings), you may be able to get around that restriction. Make sure nothing damages the wood, in case your landlord would object to that.

That's exactly how I had planned it.