Obviously I am to too stupid to understand WL's symbolic calculations.
I try to explain my question stepwise:
After a ClearAll["Global`*"] I start with some primitive examples:
In[187]:= Simplify[Log[a] - Log[b]]
Out[187]= Log[a] - Log[b]
In[188]:= Simplify[Log[a] - Log[b], Reals]
Out[188]= Log[a] - Log[b]
In[189]:= Simplify[Log[a] - Log[b], b > 0]
Out[189]= Log[a/b]
Only the last one works. But I have to know, that there is some "-Log[b]" within my expression.
Actually I have some unknown expression exp resulting from some former calculations and I have no idea what that expression will be. Say:
In[190]:= exp = Assuming[Reals, Integrate[1/(m x + c), {x, s0, s}]]
Out[190]= ConditionalExpression[(Log[1 + (m s)/c] - Log[1 + (m s0)/c])/m,
s0 > 0 && Re[s] > s0 && s == Re[s]]
BTW: I have no idea, why I get a ConditionalExpression telling something about real parts although I restricted everything to Reals. But this is another problem.
For the moment I just take the first part of exp (which is nonsense in a general case):
In[191]:= exp1 = First[exp]
Out[191]= (Log[1 + (m s)/c] - Log[1 + (m s0)/c])/m
... and Simplify it. Which doesn't change anything of course. Unlike the above primitive a,b-example I do not know the structure of my expression exp1 and hence cannot assume anything apart from being Reals:
In[192]:= Simplify[exp1, Reals]
Out[192]= (Log[1 + (m s)/c] - Log[1 + (m s0)/c])/m
If I knew the result already I could write:
In[193]:= Simplify[exp1, {m s + c > 0, m s0 + c > 0}]
Out[193]= Log[(c + m s)/(c + m s0)]/m
... and get the desired result. But I don't know anything about exp1.
Now I get to my real application. I have some arbitrary symbolic function v(s) and need the definite integral t(s) of 1/v(s) from s0 to s in a simple form. I know that v(s)>0 within the interval [s0,s]. But I do not know, what the structure of v(s) is.
Here a simple example with a straigt line for v(s)
(BTW: Without assuming Reals the Integrate takes an awfull long time):
In[194]:= v[s_] := m s + c;
t1 = Assuming[Reals, Integrate[1/v[x], {x, s0, s}]]
Out[195]= ConditionalExpression[(Log[1 + (m s)/c] - Log[1 + (m s0)/c])/m,
s0 > 0 && Re[s] > s0 && s == Re[s]]
If I assume everything I know, i.e. Reals and v(s)>0 I can write:
In[196]:= v[s_] := m s + c;
t2 = Assuming[{Reals, v[x] > 0}, Integrate[1/v[x], {x, s0, s}]]
Out[197]= ConditionalExpression[(Log[1 + (m s)/c] - Log[1 + (m s0)/c])/m,
s0 > 0 && Re[s] > s0 && s == Re[s]]
... but this doesn't change anything.
What I need is a result like t = Log[(c+ m s)/(c+m s0)]/m. How could I get this?