Log[17]/Log[2]   ->  Log2[17]
Log[13]/Log[10]  ->  Log10[13]
Log[99]/Log[11]  ->  Log[11, 99]

$PrePrint = # /. {
     Log[n_]/Log[2]  :> Defer @ Log2[n],
     Log[n_]/Log[10] :> Defer @ Log10[n],
     Log[n_]/Log[b_] :> Defer @ Log[b, n]
     } &;

MakeBoxes[Log[n_]/Log[2], fmt_]  := ToBoxes[Defer @ Log2[n], fmt]
MakeBoxes[Log[n_]/Log[10], fmt_] := ToBoxes[Defer @ Log10[n], fmt]
MakeBoxes[Log[n_]/Log[b_], fmt_] := ToBoxes[Defer @ Log[b, n], fmt]

Unprotect[Times];

Format[Log[n_]/Log[2]]  := Defer @ Log2[n]
Format[Log[n_]/Log[10]] := Defer @ Log10[n]
Format[Log[n_]/Log[b_]] := Defer @ Log[b, n]

Protect[Times];

SetSystemOptions["SimplificationOptions" -> "AutosimplifyTwoArgumentLog" -> False];

logRule = Log[x_]/Log[b_] :> Switch[b, 2, Log2[x], 10, Log10[x], _, Log[b, x]];

{Log[17]/Log[2], Log[13]/Log[10], Log[99]/Log[11]} /. logRule
   {Log2[17], Log10[13], Log[11, 99]}

Expand[(Log[17] + Log[4])/Log[10]] /. logRule
   Log10[4] + Log10[17]