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Expansion of an exponential expression


As a result of some calculations I have
(a snipet of a more complex expression)

anlog= Log[ ((2 + Sqrt[1 + 2*h])^(-1))^(t/h)   ]

I would like to simplify this to 

anlog= (t/h) Log[((2 + Sqrt[1 + 2*h])^(-1))   ]

so I can then take the Taylor series in h.
But FullSimplify[anlog,h>0]  doesnt do it.
Do i need a ComplexityFunction?


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