ListLogLinearPlot logarithmic axis tickmarks

You can create some custom ticks that work the way you want. There might be a better way than Superscript, but I couldn't work out how to get ScientificForm or NumberForm to just show the exponent rather than 1 x 10^4 etc.

myTicks = Table[{10^i, Superscript[10, i]}, {i, -20, 15}]

Notice I've changed the way the a variable enters into the Manipulate, as it will give you more chance to explore some of the lower-exponent values.

Manipulate[
 ListLogLinearPlot[{Table[{10^a s[T], T}, {T, 0, 1000}], 
   Table[{10^a t[T], T}, {T, 0, 1000}]}, GridLines -> None, 
  Ticks -> {myTicks, Automatic}, PlotStyle -> {Thickness[0.005]}, 
  Joined -> True, PlotRange -> Automatic], {a, -8, 8}]

One can adapt this answer to a smilar question to define a number format as follows:

 exponentForm[num_?NumericQ] := 
 ToString@NumberForm[N@#1, ExponentFunction -> (# &), NumberFormat -> (#3 &)] &@num

Applied to a portion of Verbeia'a tick list, this function gives

You can use it to define custom ticks with exponent labels:

  myExpTicks = Table[{10^i, exponentForm[10^i]}, {i, -20, 15}];

And, finally use it in your Manipulate as

 Manipulate[
 ListLogLinearPlot[{Table[{10^a s[T], T}, {T, 0, 1000}], 
 Table[{10^a t[T], T}, {T, 0, 1000}]}, GridLines -> None, 
 Ticks -> {myExpTicks, Automatic}, PlotStyle -> {Thickness[0.005]}, 
 Joined -> True, PlotRange -> Automatic], {a, -8, 8}]

to get

You can create custom ticks and use ListPlot :

myTicks[lower_, upper_, step_] := Flatten[Module[{Local = #, LocalTicks}, 
 LocalTicks = Log[10, Range[10^(# - 1), 10^#, (10^# - 10^(# - 1))/(10 - 1)]]; 
 Join[{#, ""} & /@ Most[LocalTicks], {{Last[LocalTicks], Power[10, 
 ToString[Last[LocalTicks]]] }}]] & /@ Range[lower, upper, step], 1]

Manipulate[ListPlot[{Table[{Log[10, a s[T]], T}, {T, 0, 1000}], Table[{Log[10, 
 a t[T]], T}, {T, 0, 1000}]}, GridLines -> None, PlotStyle -> {Thickness[0.005]}, 
 Joined -> True, PlotRange -> Automatic, Ticks -> {myTicks[lower, upper, step],
  Automatic}, AxesOrigin -> {lower, 0}], {{lower, -20, "lower"}},
  {{upper, 10, "upper"}}, {{step, 1, "step"}}, Delimiter, {a, 10^-8, 10^8}]