The function #f(x)=5^x# has horizontal asymptote #y=0# because as #x to -oo#, #f(x) to 0# and no other asymptotes.
#h(x)=5^x-2# is a horizontal shift of #f(x)# by 2 units to the right, so the range is not impacted. That means the horizontal asymptote is still #y=0#. The function has no other asymptotes.