Number format in Legend Labels

Question

I would like to use the usual (new and nice) BarLegend[] with something like

BarLegend[{"LakeColors", 1/10000 {-2, 2}}]

which looks like

I don't like the numbers, i would like to use ScientificForm[] on them, hence get $0.0002 = 2\times10^{-4}$ or even better - get just $2$ at the label and $\times 10^{-4}$ in the bottom right or something like that.

Is there a way to obtain ScientificForm[] for these labels? I searched the Documentation of the new ~Legends and haven't found anything. And the very best would be, to be able to specify the number of digits used (where i would like to have 3, e.g. $2.00\times10^{-4}$, so best would be ScientificForm[#,3] & to be applied to every number of the Legend.

Update

Surprisingly - following the approach of @Nasser M. Abbasi the two lines

f[x_] := ScientificForm[x, 2];
BarLegend[{"LakeColors", 1/10000 {-2, 2}}, LegendFunction -> f]

produce a Legend like

Though the second argument of ScientificForm[] gets ignored. Any further ideas why that does happen?

Update #2

Changing f to f[x_] := ScientificForm[N[{x} /. {DirectedInfinity -> Identity}], 2]; actually does change the number of digits, but returns the BarLegend in an Array and produces errors, that (1.,4.} is not a List of positive Integers (though they look quite Integer to me). Why there is an DirectInfinity approaching, I haven't found out yet, without the replacement, the 1. is a DirectedInfinity[1.]

Answer 1

Thanks to Nasser M. Abbasi i found a way. To change the Display. The function that you can provide for any ~Legend via LegendFunction wraps the complete ~Legend[] into anything. And he mentioned, that NumberForm encapsulates the numbers. So why not replace them (delayed)?

Version 1: Scientific Notation at every Label

Hence choosing

f[x_] := x /. {NumberForm[y_, {w_, z_}] :> ScientificForm[y, 2]}

to replace any NumberForm in the Legend by ScientificForm (using my preferred two digits) we may call

BarLegend[{"LakeColors", N[1/10000 {-2, 2}]}, LegendFunction -> f]

and obtain

which is one of the desired results.

Version 2: Extract exponent into the label

The first version might seem a little crowded for some Legends, so a nicer way would be to extract that Exponent from the ScientificFormat and change back to NumberFormat internally. While the first might also work, if the Range is implicitly given, this requires to know

{min, max} = 1/10000 {-2,10};

beforehand. Notice, that in this range, the Exponents of max ($10^{-3}$) and min ($-2\times10^{-4}$) differ.

We calculate

exp = Log[10, Max[Abs[min], Abs[max]]]

which is $-3$ here. And change the approach from before to

f[x_] := x /. {NumberForm[y_, {w_, z_}] :> 
NumberForm[PaddedForm[y/(10^exp), {1, 2}], 2]}

That way we can add a Label to the ~Legend, for containing this exponent, for example

BarLegend[{"LakeColors", {-min, max}}, LegendFunction -> f,
  LegendLabel -> Placed[DisplayForm[SuperscriptBox[ToString[" \[Times] 10"], exp]], Bottom]]

Which I think is nicer than the first version, though you have to know the range of values beforehand. The space before \[Times] just places the label a little further to the right, such that it's bounded at the right hand side.