How can I simplify $\log(512)$ to $9\log(2)$?

Question

Can you explain me about the result of Integrate.

\!\(\*SubsuperscriptBox[\(\[Integral]\), \(1\), \(3\)]\(\*SubsuperscriptBox[\(\[Integral]\), 
    \(1\), \(3\)]\*FractionBox[\(9  v\), \(v + u\ v\)] \[DifferentialD]u \[DifferentialD]v\)\)
(*
    2 log(512)
*)

% // Simplify
(*
    2 log(512)
*)

% // FullSimplify
(*
    18 log(2)
*)

However, I cannot FullSimplify this result

FullSimplify[Log[512]]
(*
    log(512)
*)

Log[512] // N
(*
    6.23832
*)

FullSimplify[2 Log[512]] // N
(*
    12.4766
*)

So question $1$ is how to simplify Log[512] and why it doesn't simplify to 9Log[2] by default.

9 Log[2] // N
(*
    6.23832
*)

When I'm integrating by hand, I collect 9 or 18 easily, so I like the result 18 log(2) , not 2 log(512), so I wonder why Mathematica gives this result?

Answer 1

Mathematica does not consider 9 Log[2] to be "simpler" than Log[512]. The full default ComplexityFunction is not disclosed (the one in the documentation is not entirely equivalent IIRC), but a good first-order approximation is often LeafCount

LeafCount /@ {Log[512], 9 Log[2]}

{2, 4}

If you provide a ComplexityFunction that uses a different metric by which 9 Log[2] has a lower score than Log[512], such as the largest integer that appears in the expression, it will reduce as you desire:

FullSimplify[Log[512], ComplexityFunction -> (Max @ Cases[#, _Integer, {0, -1}] &)]

9 Log[2]

Answer 2

This kind of transformation Log[512] -> 9 Log[2] is not a simplification of the underlying expression but rather expanding it, there is PowerExpand which appears to be the simplest approach:

PowerExpand @ Log @ 512

9 Log[2]

We can see that Log[512] is simpler than 9 Log[2], to demonstrate it we can use TreeForm:

GraphicsRow[ TreeForm /@ {9 Log[2], Log[512]}]

When we are trying to transform adequate expressions with FullSimplify LeafCount only approximates ComplexityFunction, however in general it can't explain specific simplifications. Moreover ComplexityFunction works in a different way in Mathematica 9 in comparison to earlier versions, see e.g. FullSimplify does not work on this expression with no unknowns discussing how one can deal with similar problems.
Having said that I find simpler using the TransformationsFunctions option of FullSimplify rather than searching for appropriate ComplexityFunction, namely here we would use PowerExpand:

FullSimplify[ Log[512], TransformationFunctions -> PowerExpand]

9 Log[2]

Now FullSimlify uses only PowerExpand while (see the documentation):

TransformationFunctions -> {Automatic, $f_1$, $f_2$, $\ldots$ } uses built-in transformation functions together with the functions $f_i$.