python
diff --git a/‎Doc/library/cmath.rst‎
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diff --git a/‎Doc/library/math.rst‎
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diff --git a/‎Lib/test/test_math.py‎
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diff --git a/‎Misc/NEWS.d/next/Library/2023-01-16-10-42-58.gh-issue-89381.lM2WL0.rst‎
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diff --git a/‎Modules/clinic/cmathmodule.c.h‎
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diff --git a/‎Modules/clinic/mathmodule.c.h‎
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diff --git a/‎Modules/cmathmodule.c‎
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diff --git a/‎Modules/mathmodule.c‎
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@@ -89,7 +89,7 @@ Power and logarithmic functions
    logarithms.
 
 
-.. function:: log(x[, base])
+.. function:: log(x, base=None)
 
    Returns the logarithm of *x* to the given *base*. If the *base* is not
    specified, returns the natural logarithm of *x*. There is one branch cut, from 0
 
@@ -393,13 +393,12 @@ Power and logarithmic functions
    .. versionadded:: 3.2
 
 
-.. function:: log(x[, base])
+.. function:: log(x, base=None)
 
-   With one argument, return the natural logarithm of *x* (to base *e*).
-
-   With two arguments, return the logarithm of *x* to the given *base*,
-   calculated as ``log(x)/log(base)``.
+   Return the logarithm of *x* to the given *base*.
 
+   If the *base* is not specified, returns the natural
+   logarithm (base *e*) of *x*.
 
 .. function:: log1p(x)
 
 
@@ -1146,6 +1146,7 @@ def testLog(self):
         self.ftest('log(1/e)', math.log(1/math.e), -1)
         self.ftest('log(1)', math.log(1), 0)
         self.ftest('log(e)', math.log(math.e), 1)
+        self.ftest('log(e, None)', math.log(math.e, None), 1)
         self.ftest('log(32,2)', math.log(32,2), 5)
         self.ftest('log(10**40, 10)', math.log(10**40, 10), 40)
         self.ftest('log(10**40, 10**20)', math.log(10**40, 10**20), 2)
 
@@ -0,0 +1 @@
+:func:`~math.log` and :func:`~cmath.log` support default base=None values.
@@ -952,23 +952,24 @@ cmath_tanh_impl(PyObject *module, Py_complex z)
 cmath.log
 
     z as x: Py_complex
-    base as y_obj: object = NULL
+    base as y_obj: object = None
     /
 
 log(z[, base]) -> the logarithm of z to the given base.
 
-If the base not specified, returns the natural logarithm (base e) of z.
+If the base is not specified or is None, returns the
+natural logarithm (base e) of z.
 [clinic start generated code]*/
 
 static PyObject *
 cmath_log_impl(PyObject *module, Py_complex x, PyObject *y_obj)
-/*[clinic end generated code: output=4effdb7d258e0d94 input=230ed3a71ecd000a]*/
+/*[clinic end generated code: output=4effdb7d258e0d94 input=e7db51859ebf70bf]*/
 {
     Py_complex y;
 
     errno = 0;
     x = c_log(x);
-    if (y_obj != NULL) {
+    if (y_obj != Py_None) {
         y = PyComplex_AsCComplex(y_obj);
         if (PyErr_Occurred()) {
             return NULL;
 
@@ -2366,26 +2366,24 @@ loghelper(PyObject* arg, double (*func)(double))
 math.log
 
     x:    object
-    [
-    base: object(c_default="NULL") = math.e
-    ]
+    base: object = None
     /
 
 Return the logarithm of x to the given base.
 
-If the base not specified, returns the natural logarithm (base e) of x.
+If the base is not specified or is None, returns the natural
+logarithm (base e) of x.
 [clinic start generated code]*/
 
 static PyObject *
-math_log_impl(PyObject *module, PyObject *x, int group_right_1,
-              PyObject *base)
-/*[clinic end generated code: output=7b5a39e526b73fc9 input=0f62d5726cbfebbd]*/
+math_log_impl(PyObject *module, PyObject *x, PyObject *base)
+/*[clinic end generated code: output=1dead263cbb1e854 input=ef032cc9837943e1]*/
 {
     PyObject *num, *den;
     PyObject *ans;
 
     num = loghelper(x, m_log);
-    if (num == NULL || base == NULL)
+    if (num == NULL || base == Py_None)
         return num;
 
     den = loghelper(base, m_log);
Original file line number	Diff line number	Diff line change
`@@ -0,0 +1 @@`
	`1`	+:func:`~math.log` and :func:`~cmath.log` support default base=None values.