Graph y=ln(x+1)

This is the same as the graph of y=ln(x), shifted LEFT by 1 unit.

y=ln(x) has a vertical asymptote at x=0, and then an x-intercept at (1,0).
Both of these things shift left 1: The Vertical Asymptote is now at x=-1, and the x-intercept is (0,0)