what exact procedure is taken to increase the log probability by log(2) and return modified percentages?

The simplest way is to use odds ratios instead of log probability. 5% is 1:19. Multiply that by 2:1 and you get 2:19 which corresponds to 9.52%. If it's close to 100%, you get close to half the probability of failure. If it's close to 0%, you get close to double the probability of success.

This can be done with dice by using a virtual d21. You can do that by rolling a higher-numbered die and re-rolling if you pass 21. Since the next die up is d100, you can combine two dice to get d24 or d30 the same way you combine two d10s to get a d100. Alternately, use a computer or a graphing calculator instead of a die, and you can have it give whatever probabilities you want.