WE've used a little canon powershot for years. IT was probably more expensive than you budget--but they are cheaper now. Very lightweight, has image stabilization, and a decent zoom. ANd I you leave the display off most of the time, the battery will last well longer than a one-week backpacking trip!

you can the photos it took on our website---they were all done with the canon PS780...
_________________________
Check our our website: http://www.backpackthesierra.com/

Or just read a good mystery novel set in the Sierra; https://www.amazon.com/Danger-Falling-Rocks-Paul-Wagner/dp/0984884963