diff --git a/src/sage/rings/integer.pyx b/src/sage/rings/integer.pyx
index 7d0e6f34464..44523fed015 100644
--- a/src/sage/rings/integer.pyx
+++ b/src/sage/rings/integer.pyx
@@ -2851,16 +2851,26 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
             sage: ZZ(8).log(int(2))
             3
 
+        Check that negative bases yield complex logarithms (:issue:`39959`)::
+
+            sage: 8.log(-2)
+            3*log(2)/(I*pi + log(2))
+            sage: (-10).log(prec=53)
+            2.30258509299405 + 3.14159265358979*I
+
+        Check that zero base  yield complex logarithms (:issue:`39959`)::
+
+            sage: 8.log(0)
+            0
+
         TESTS::
 
             sage: (-2).log(3)                                                           # needs sage.symbolic
             (I*pi + log(2))/log(3)
         """
         cdef int self_sgn
-        if m is not None and m <= 0:
-            raise ValueError("log base must be positive")
         self_sgn = mpz_sgn(self.value)
-        if self_sgn < 0 and prec is None:
+        if (self_sgn < 0 or m is not None and m<=0) and prec is None:
             from sage.symbolic.ring import SR
             return SR(self).log(m)
         if prec: