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I tried:
sage: p = 11
sage: K = Qp( p )
sage: R.<x> = PolynomialRing( K )
sage: L.<b> = K.extension( x^p - p )
sage: b
b + O(b^221)
sage: b.norm()
11 + O(11^21)
sage: -log( 1-b ) + O(b^12)
1 + b + 6*b^2 + 4*b^3 + 3*b^4 + 9*b^5 + 2*b^6 + 8*b^7 + 7*b^8 + 5*b^9 + 10*b^10 + 6*b^11 + O(b^12)
# But...
# sage: var( 'y' );
# sage: solve( y^2 - b*y + 7 == 0 , y ) # --> error
