f(x) = ln (x+4) -2
Change f(x) to y
y=ln (x+4) -2
Inverse x and y
x=ln(y+4) -2
Solve for y, result will be inverse.
x+2=ln(y+4)
convert to exponential form
e^(x+2) = y+4
y=e^(x+2) -4
This is the inverse.
f^-1(x) = e^(x+2) -4
Looking at the above function, it is (e^x) shifted to the left by two unit, being brought down (4) units. The only important part of this function is the down-shift of (4), because this represents the horizontal asymptote.
y = -4 is the horizontal asymptote, which means
for f(x) =ln(x+4) -2
x=-4 is a vertical asymptotes.