Map and Apply a function on a nested list

Since you seem to be relatively new to Mathematica, and unfamiliar with all its special syntax (/@, @@@ etc), I would normally recommend Artes' second answer:

{#1, Log[#2]} & @@@ {{1, 2}, {4, 2}, {6, 4}}

Which can also be written

Apply[{#1, Log[#2]} &, testdata, {1}]

where testdata = {{1, 2}, {4, 2}, {6, 4}}

Some alternative ways of getting the same answer include:

MapThread[{#1, Log[#2]} &, Transpose@testdata]

and (I think this one is quite cool)

Inner[#1[#2] &, {# &, Log[#] &}, Transpose@testdata, List]

MapAt and deeply nested lists generalization

Another way to do this:

MapAt[Log, #, 2] & /@ {{1,2}, {4,2}, {6,4}}

{{1, Log[2]}, {4, Log[2]}, {6, Log[4]}}

Which is useful if we target a specific element inside every element of a deeply nested list:

data = Table[{k, {k, {{k}}}}, {k, 2, 5}]

{{2, {2, {{2}}}}, {3, {3, {{3}}}}, {4, {4, {{4}}}}, {5, {5, {{5}}}}}

MapAt[Log, #, {2, 2}] & /@ data

{{2,{2,{{Log[2]}}}}, {3,{3,{{Log[3]}}}}, {4,{4,{{Log[4]}}}}, {5,{5,{{Log[5]}}}}}

You can use ReplaceAll i.e.

{{1, 2}, {4, 2}, {6, 4}} /. {a_, b_} -> {a, Log[b]}

{{1, Log[2]}, {4, Log[2]}, {6, Log[4]}}

or

{#1, Log[#2]} & @@@ {{1, 2}, {4, 2}, {6, 4}}

i.e. Apply the function {#1, Log[#2]} & on the first level of the expression.