changing intercept logistic regression in R I did logistic regression and got an intercept of -14 and a beta_1 of 0.17.
My equation is: -14 + 0.17*x = y
How can I calculate the change of probability if x increases by 0.1, assuming a original probability of 20%?
Can I change beta_0 to log(0.2/0.8) and then add 0.1*0.17?

You're on the right track. You need to do the calculation you suggested, then convert back to the probability scale and subtract 0.2 (the baseline probability).
plogis(qlogis(0.2) + 0.17*0.1) - 0.2
## [1] 0.002733877

qlogis() is the logit (log-odds) transformation (i.e. log(0.2/0.8) = qlogis(0.2)); plogis() is the logistic, or inverse-logit transformation
If the starting probability were closer to 0.5 you could use the rule of thumb that the inverse-logit/logistic curve is approximately linear with slope beta/4 in the range (0.3, 0.7); in this case 0.17/4*0.1 gives 0.00425, which is the right order of magnitude but off by about 50% ... (for a starting probability of 0.3 it's only off by 30%)
