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RE: This should evaluate to zero


jf,

Mathematica doesn't simplify because there are multiple solutions.

You could do this..

ArcSinh[1] - Log[3363 + 2378*Sqrt[2]]/10 // N
0.

But if you want to work with exact numbers do the following...

ArcSinh[1] - Log[3363 + 2378*Sqrt[2]]/10
TrigToExp[%]
% /. Log[a_] + b_*Log[c_] -> Log[a*c^b]

giving

ArcSinh[1] - (1/10)*Log[3363 + 2378*Sqrt[2]]
Log[1 + Sqrt[2]] - (1/10)*Log[3363 + 2378*Sqrt[2]]
Log[(1 + Sqrt[2])/(3363 + 2378*Sqrt[2])^(1/10)]

Then it is clear that we have to find the 10th root of the denominator. It
turns out there are two real roots and only one positive real root. We can
select that and substitute with the following statement...

sols = First[Select[Solve[z^10 == 3363 + 2378*Sqrt[2]],
    With[{x = #1[[1,2]]}, Im[x] == 0 && Re[x] > 0] & ]]
Log[(1 + Sqrt[2])/z] /. sols

{z -> 1 + Sqrt[2]}
0

David Park
djmp at earthlink.net
http://home.earthlink.net/~djmp/


From: jf alcover [mailto:jf.alcover at bdpme.fr]
To: mathgroup at smc.vnet.net

Bonjour,
Methinks that the following expression should evaluate to zero,
but it does not, even with FullSimplify :
    ArcSinh[1]-Log[3363+2378*Sqrt[2]]/10
Could anyone explain ?



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