Think about the graph of y = log x.
This has a vertical asymptote at x = 0 because log 0 is undefined (und). Why? What exponent could you put onto 10 to make 0? `10^x=0` ? Similarly, log x is undefined for negative numbers.
Now, compare log x to log (x-1). Maybe a little table will help:
x 0 1 2 3
log x und 0 ~0.3 ~0.5
log x-1 und und 0 ~0.3
So it appears that all values got shifted to the right by one unit...
In fact, the graph of log(x-1) looks just like the graph of log(x) except shifted right one unit.
So if the vertical asymptote in y = log x occurs at x=0, just shift it over one unit to the right for y = log (x-1).