Display Log with desired notation in output

One can use $PrePrint and ReplaceAll to effect this:

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

Or MakeBoxes:

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]

It is also possible to use Format but in this case it requires unprotecting Times:

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];

These assignments are made to a special class of rules: FormatValues. Because these are only used in formatting this should not slow down internal operations using Times, unlike overloading UpValues or DownValues.

Each method relies on Defer to prevent an infinite recursion yet allow evaluation of output when it is given as input.

Result:

Using a trick similar to what Chip showed in this answer:

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]}

Sjoerd's example in a comment to the Wizard's answer requires some finesse:

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