This should work, using a replacement rule to replace every y value with a scaled version, and then making sure that you have a scaled tick mark function as well

plot = Plot[{x^x, Exp[x], x!}, {x, 1, 5}, 
  PlotStyle -> {Blue, Red, Green}]
Show[plot /. Line[x__] :> (({#1, Log@#2} & @@@ x) // Line),
 PlotRange -> All,
 Ticks -> {Automatic, Charting`ScaledTicks[{Log, Exp}]}]

This should work, using a replacement rule to replace every y value with a scaled version, and then making sure that you have a scaled tick mark function as well

plot = Plot[{x^x, Exp[x], x!}, {x, 1, 5}, 
  PlotStyle -> {Blue, Red, Green}]
Show[plot /. Line[x__] :> (({#1, Log@#2} & @@@ x) // Line),
 PlotRange -> All,
 Ticks -> {Automatic, Charting`ScaledTicks[{Log, Exp}]}]

Or, with OP's example more explicitly

listplot = 
  ListPlot[Transpose[{#^#, Exp[#], #!} & /@ Range[1, 5]], 
   PlotStyle -> {Darker@Blue, Darker@Red, Darker@Green}];
plot = Plot[{x^x, Exp[x], x!}, {x, 1, 5}, 
   PlotStyle -> {Blue, Red, Green}, PlotRange -> All];
combined = Show[plot, listplot]
Show[combined /.
  {Line[x__] :> (({#1, Log@#2} & @@@ x) // Line), 
   Point[x__] :> (({#1, Log@#2} & @@@ x) // Point)},
 PlotRange -> All,
 Ticks -> {Automatic, Charting`ScaledTicks[{Log, Exp}]}]