Mathematical logical Solution:

log [1/(10^10)] = -10 log(10) = -10 ->A

log [1/(10^20)] = -20 log(10) = -20 ->B

Now,

So, LHS: A + 3B + 80 + 90

=> -10 + 3(-20) + 80 + 90

=> 100 -> RHS

Hence proved! ( No negative numbers though)

Hence, A is log[1/(10^10)] and B is log[1/(10^20)]

#Puzzle_answer: 105$

Let us assume that the cost of phone case is x

Therefore, cost of phone is x+100

x+(X+100)=110

2x=10

x=5

Therefore,

Phone cost =5+100 =105

Phone cost is 105$