# Why does WL not know that Log[a]-Log[b] is Log[a/b]?

Posted 7 months ago
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 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?
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Posted 7 months ago
 (1) I am pretty sure Assuming[Reals,...] has no meaning. Reals can be used as a domain in solvers that accept such an argument. Assumptions by contrast expects explicit assumptions on stated parameters e.g. c>0.(2) That second Simplify looks fine, in that log a - log b is not equal to log a/b for arbitrary real (a,b). Log[a] - Log[b] - Log[a/b] /. {a -> 3, b -> -1} (* Out[1]= -2 I \[Pi] *) (3) For parametrized definite integrals, it generally helps to use as many assumptions as possible. Below is an example which could be modified for specific needs. v[s_] := m s + c; t1 = Integrate[1/v[x], {x, s0, s}, Assumptions -> {m > 0, c > 0, 0 < s < s0}] (* Out[416]= Log[(c + m s)/(c + m s0)]/m *) 
Posted 7 months ago
 Thanks, Daniel. Of course Log[a]-Log[b}==Log[a/b holds true only for a>0 && b>0. But this is not my problem.Your proposition (3) doesn't help since it makes assumptions about the structure of my expression. Actually I do not know anything about  Integrate[1/v[x],{x,s0,s}] apart from v[x] being real and positive within the real integration interval [s0,s]. Not even s>=s0 needs to be. Hence I think I cannot write more than the t2 in my original post.Anyway I can live with log(a)-log(b) not being resolved to log(a/b).
Posted 7 months ago
 In[3]:= Reduce[Log[a] - Log[b] == Log[a/b], {a, b}, Reals] Out[3]= a > 0 && b > 0 
Posted 7 months ago
 Ja, of course. That's known, but not my problem.
Posted 7 months ago
 A kind of roundabout way: In[1]:= Log@Exp[Log@a - Log@b] Out[1]= Log[a/b] Applied to Werners example: In[2]:= t = (Log[1 + (m s)/c] - Log[1 + (m s0)/c])/m Out[2]= (Log[1 + (m s)/c] - Log[1 + (m s0)/c])/m In[3]:= t /. Log[a_] - Log[b_] -> Log@Exp[Log@a - Log@b] Out[3]= Log[(1 + (m s)/c)/(1 + (m s0)/c)]/m 
 Your Replace-idea will certainly work and does not disturb anything if my expression does not contain loag(a)-log(b). I think it can even be simplified: In[275]:= v[s_] := m s + c; t3 = Assuming[{Reals, v[x] > 0}, Integrate[1/v[x], {x, s0, s}]] Out[276]= ConditionalExpression[(Log[1 + (m s)/c] - Log[1 + (m s0)/c])/m, s0 > 0 && Re[s] > s0 && s == Re[s]] In[277]:= t3 = Simplify[t3 /. Log[a_] - Log[b_] -> Log[a/b]] Out[277]= ConditionalExpression[Log[(c + m s)/(c + m s0)]/m, s0 > 0 && Re[s] > s0 && s == Re[s]] 
 BTW: To avoid the ConditionalExpression from Integrate, which is useless in my case, one can suppress them with option GenerateConditions. Instead auf writing: In[359]:= v[s_] := m s + c; t5 = Assuming[Reals, Integrate[1/v[x], {x, s0, s}]] Out[360]= ConditionalExpression[(Log[1 + (m s)/c] - Log[1 + (m s0)/c])/m, s0 > 0 && Re[s] > s0 && s == Re[s]] ... one can write: In[361]:= v[s_] := m s + c; t6 = Assuming[Reals, Integrate[1/v[x], {x, s0, s}, GenerateConditions -> False]] Out[362]= (Log[1 + (m s)/c] - Log[1 + (m s0)/c])/m `