34212: tests and docs

src/sage/rings/number_field/number_field_element.pyx

@@ -61,6 +61,7 @@ import sage.rings.rational_field
 import sage.rings.rational
 import sage.rings.integer_ring
 import sage.rings.integer
+from sage.rings.real_mpfr import RealField
 
 from sage.arith.power cimport generic_power
 from sage.rings.real_mpfi cimport RealIntervalFieldElement
@@ -4123,7 +4124,26 @@ cdef class NumberFieldElement(FieldElement):
 
         The logarithmic embedding is a tuple of real embeddings and conjugate
         pairs of compelx embeddings.
+
         It is defined under Definition 4.9.6 in [Cohen1993]_.
+
+        INPUT:
+
+        - ``x`` -- a rational number.
+        - ``prec`` -- desired floating point precision.
+
+        OUTPUT:
+
+        - a tuple of real numbers.
+
+        EXAMPLES::
+
+            sage: K.<a> = NumberField(x^3 + 5)
+            sage: f = a^2 - 3
+            sage: f.log_embedding(10)
+            (2.30258509299405)
+            sage: f.log_embedding(0)
+            (-1)
         """
         K = self.base_ring()
         K_embeddings = K.places(prec)