E_ion = +62/z log ([E_out]/[E_in])
z is the valency, E is the membrane potential at which there is no net movement of ion, if there were only that ion. The more open channels a specific ion species has, the closer the membrane potential (Vm) is to the E_ion (usually in millivolts).
If there are 10x more K+ ions inside, E_K+~-62 mV. Now since the E_Na+ is usually some positive number (let's say 62mV), there is a driving force of ~124mV for Na to go into the cell... but all the Na channels are closed! Since the cell is resting at Vm~ -60-80 mV there is little incentive for K+ ions to leave: the cell is already close to E_K+. Vm is usually measured using electrodes but it can be predicted with the Goldmann eq which takes into account the relative permeability and E_ion of each ion and their channels.
K does not cause the inside to be negative. Your cells pump K in and Na out and keeps them out, generating a negative membrane potential (closer to E_K+ since there are more open K+ leak channels).