# Simplify expressions with Log

How can I get Mathematica to simplify the following expression

n Log[a] + m Log[b] - m Log[a + b] - n Log[a + b]


into

Log[ a^n b^m (a + b)^(-m - n)] ?


I've tried various methods without any luck including: -

FullSimplify[ n Log[a] + m Log[b] - m Log[a + b] - n Log[a + b], { a + b > 0 } ]


Perhaps I'm not including enough assumptions or Mathematica doesn't consider this to be a simplification ? It would be nice to have a solution that doesn't require pattern matching.

• In general Log[a^n] does not equal n*Log[a]. Take for instance Log[(-1)^2]. However, MMA knows that Assuming[{n > 0, m > 0, a > 0, b > 0}, FullSimplify[n Log[a] + m Log[b] == Log[a^n b^m]]] is True. So, these assumptions should be sufficient, but I cannot let it make the simplification itself. Apr 4, 2013 at 22:23
• This Q&A seems to be a bit overlooked. The OP there wants to do things the other way around, i.e. go from a Log of a product to the sum of Logs. It is a subjective matter what a simplification is, so I like Artes answer here below. The OP in the link uses something similar. The answer in the link by wolfies just points to proprietary software, but can possibly provide some (professional) context. The answer in the link by Andre also seems great. Apr 5, 2013 at 9:39
• Only for completeness: You can perform the reverse operation (expanding the Log[] function) via PowerExpand[].
– loki
Dec 14, 2017 at 13:40

Let us introduce the function to transform the logarithm:

    collectLog[expr_] := Module[{rule1, rule2, a, b, x},
rule1 = Log[a_] + Log[b_] -> Log[a*b];
rule2 = x_*Log[a_] -> Log[a^x];
(expr /. rule1) /. rule2 /. rule1 /. rule2
];


expr = (n Log[a] + m Log[b] - m Log[a + b] - n Log[a + b]);


Let us first simplify it, and then apply to it the collectLog function:

    expr2 = Simplify[expr, {a > 0, b > 0},
TransformationFunctions -> {Automatic, ComplexExpand}] //
collectLog


The result is

Log[a^n b^m] + Log[(a + b)^(-m - n)]

Let us apply the collectLog once more:

expr2 // collectLog


The result is:

Log[a^n b^m (a + b)^(-m - n)]


Done.

To answer the recent question of bszd: if a function with multiple Logs may be designed.

It can be done in a more simple way. If one has a lengthily expression with logarithms of the sort that might be simplified by collection, the function Nest may do the job:

 Nest[collectLog, expr, Length[expr]]


Log[a^n b^m (a + b)^(-m - n)]


If it is only a part of expression that, however, contains multiple logarithms to be collected, the function

collectAllLog[expr_] := Nest[collectLog, expr, Length[expr]];


may be mapped onto this part.

Finally, to complete this one may need to do the opposite operation: to expand the logarithmic expression. One way to do this would be to use the following function:

    expandLog[expr_] := Module[{rule1, rule2, a, b, x},
rule1 = Log[a_*b_] -> Log[a] + Log[b];
rule2 = Log[a_^x_] -> x*Log[a];
(expr /. rule1) /. rule2
];


and

expandAllLog[expr_] := Nest[expandLog, expr, Depth[expr]]


For example,

expandAllLog[Log[a^n b^m (a + b)^(-m - n)]]


yields

n Log[a] + m Log[b] + (-m - n) Log[a + b]


as expected.

• This looks very good. Is there any easy way I can allow multiple arguments for rule1, i.e. rule1 = Log[a_] + Log[b_] + Log[]... -> Log[a*b*...]? Apr 6, 2013 at 14:27
• @bsdz, Please have a look at the part I added to the answer. Apr 8, 2013 at 7:38
• Perfect - that's exactly what I was looking for with the added benefit it can be applied in other situations too. Thanks! Apr 8, 2013 at 21:42
• I am wondering how can I do the same replacement rules with functions sitting in front ? $e^{-i x}\log(a) + e^{ -i x} \log(b) \rightarrow e^{-i x}\log(a*b)$ May 6, 2019 at 13:56
• @ Jaswin There are several ways. 1) You may introduce a rule: rule = u_*Log[v_] + u_*Log[w_] :> u*Log[v*w] and apply it to your expression; expr = E^(-I*x)*Log[a] + E^(-I*x)*Log[b] as follows: expr /. rule. 2) Otherwise, you may first factor the expression and then apply to its part the collectLog function: MapAt[collectLog, expr // Factor, 2]. May 6, 2019 at 14:31

I know, I know: Now someone will ask why. Anyway:

FullSimplify@Log@Exp[n Log[a] + m Log[b] - m Log[a + b] - n Log[a + b]]

(* Log[a^n b^m (a + b)^(-m - n)] *)

• Short and sweet. I mean that. Not like when my HS track coach would say it (which roughly translated to "I'm trying to make you puke this afternoon"). Apr 4, 2013 at 23:01
• That's great. Shall test it tomorrow. Thank you Apr 4, 2013 at 23:04
• Simplify works, too (in place of FullSimplify). Apr 5, 2013 at 0:19
• Why does this work? :^) Apr 5, 2013 at 8:08
• @bsdz It was a joke ... Apr 6, 2013 at 11:41

Well, although late, here's an answer using ReplaceRepeated (//.).

Let's define two replacement rules to take us back and forth.

logrule = {Log[x_] + Log[y_] :> Log[x y], n_ Log[x_] :> Log[x^n]}

revlogrule = {Log[x_ y_] :> Log[x] + Log[y], Log[x_^n_] :> n Log[x]}


expr = n Log[a] + m Log[b] - m Log[a + b] - n Log[a + b]


using the logrule we can simplify your expression:

expr //. logrule


Which gives:

Log[a^n b^m (a + b)^(-m - n)]

Now let's go back to the original expression using revlogrule

Log[a^n b^m (a + b)^(-m - n)] //. revlogrule // Expand


n Log[a] + m Log[b] - m Log[a + b] - n Log[a + b]

EDIT

You can also use FullSimplify with TransformationFunctions as follows. First define the transformation you desire to be applied:

tfunc[x_] := x /. logrule


Then:

FullSimplify[expr, TransformationFunctions -> {Automatic, tfunc}]


Which gives as before:

Log[a^n b^m (a + b)^(-m - n)]

• I don't believe you need the rule Log[x_] - Log[y_] :> Log[x/y] as I think this should be handled by Log[x_] + Log[y_] :> Log[x y]. Likewise Log[x_/y_] :> Log[x] - Log[y]. There can be surprises but I don't think that particular case applies here. (+1) Sep 23, 2013 at 11:30
• @Mr.Wizard, Ah yes, fixed. Thanks. Sep 23, 2013 at 11:51
• By the way, if this account is yours you might request a merge. Sep 24, 2013 at 14:57
• @Mr.Wizard. Thanks a lot!. I've been looking for a way to do that. Sep 24, 2013 at 18:24

logList = n Log[a] + m Log[b] - m Log[a + b] - n m Log[a + b]
- n Log[a + b] - Log[ArcSin[a + b n]] + z Log[z];

mahmoh = Log[Times @@ (logList /. Times[x___, Log[y___]] :>
Power[y,Times[x]] /. Log[x__] :> Power[x, 1])]

Out[376]=Log[(a^n b^m (a + b)^(-m - n - m n) z^z)/ArcSin[a + b n]]


All is fine as long as you dont use products of Log and here is a little Test

logList /. {a -> 2.3, b -> 4.3, n -> .4, m -> .7, z -> 3}
Out[377]= 1.09137 + 0.921334 I
mahmoh /. {a -> 2.3, b -> 4.3, n -> .4, m -> .7, z -> 3}
Out[378]= 1.09137 + 0.921334 I


And using the nice package MaTeX you get this little nice picture.

MaTeX[mahmoh, Magnification -> GoldenRatio]


I did not give it a very carefull thought so maybe you can find situation when it will not work. any suggestion would be more than welcome.

As for the original question the answer is

Log[a^n b^m (a + b)^(-m - n)]