Q: My Macbook air 2013 keyboard backlight  went off

It may simply be a loose contact that needs to be pegged in again. I had a similar problem- my MacBook Pro does not like bike rides over cobbled stones.

The Apple Store can check, if a wire inside the keyboard needs reconnecting.

If you haven't already done so, please try this simple step first.

If the keyboard on your portable Mac can be backlit, you can set it so that the keys are visible in low light conditions. You can also automatically turn off backlighting when the keyboard isn’t being used.

1. Choose Apple menu > System Preferences, then click accessibility > keyboards .
2. To enable backlighting, select “Adjust keyboard brightness in low light.”When the checkbox is selected, backlighting turns on automatically. When the checkbox is unselected, the F5 and F6 keys control the backlighting.

The light sensors are located in your computer’s speaker grilles, so make sure the grilles are not covered when you use your Mac in low light conditions.

if its not a logical board problem

Why are you suspecting a logic board problem?  Are there any other indicators of such a problem, like frequent crashes or kernel panics?

Make a "Genius" appointment at an Apple Store, or go to another authorized service provider. You may have to leave the machine there for several days.

Back up all data on the internal drive(s) before you hand over your computer to anyone. There are ways to back up a computer that isn't fully functional—ask if you need guidance.

If privacy is a concern, erase the data partition(s) with the option to write zeros* (do this only if you have at least two complete, independent backups, and you know how to restore to an empty drive from any of them.) Don’t erase the recovery partition, if present.

Keep your confidential data secure during hardware repair.

Apple recommends that you deauthorize a device in the iTunes Store before having it serviced.

*An SSD doesn't need to be zeroed.