In[1]:= a = RandomReal[{-1, 1}, 100000];

In[2]:= b = RandomReal[{-1, 1}, 100000];

In[3]:= (* notice that *)
Log[Abs[{x1, x2} - {y1, y2}]]

Out[3]= {Log[Abs[x1 - y1]], Log[Abs[x2 - y2]]}

In[4]:= Timing[er = Transpose@{Range[Length[a]], Log[Abs[a - b]]};]

Out[4]= {0.0156001, Null}

In[5]:= er[[1 ;; 5]]

Out[5]= {{1, -0.0678263}, {2, -0.753793}, {3, -0.00108664}, {4, 
  0.419092}, {5, -2.70754}}

Also, it is fastest to give Mathematica real values:

In[7]:= zeros = Table[0, {1000000}];

In[9]:= Timing[Log[zeros];]

Out[9]= {0.374402, Null}

In[10]:= ones = Table[1, {1000000}];

In[11]:= Timing[Log[ones];]

Out[11]= {0.312002, Null}

In[12]:= realOnes = Table[1., {1000000}];

In[13]:= Timing[Log[realOnes];]

Out[13]= {0., Null}

In[1]:= $Version

Out[1]= "10.3.1 for Microsoft Windows (64-bit) (December 9, 2015)"

In[2]:= a = RandomReal[{-1, 1}, 100000];
b = RandomReal[{-1, 1}, 100000];
t = Range[Length[a]];
xa = Table[{t[[i]], a[[i]]}, {i, 1, Length[a]}];
xb = Table[{t[[i]], b[[i]]}, {i, 1, Length[b]}];

In[7]:= Timing[
 er = Table[{xa[[i, 1]], Log@Abs[xa[[i, 2]] - xb[[i, 2]]]}, {i, 1, 
     Length@xa}];]

Out[7]= {0.5625, Null}

In[8]:= Timing[er2 = {t, Log[Abs[a - b]]}\[Transpose];]

Out[8]= {0.015625, Null}

In[9]:= er == er2

Out[9]= True

Regards, Michael