diff --git a/Doc/library/cmath.rst b/Doc/library/cmath.rst
index 5ed7a09b3e9db2..c10c967f408a3f 100644
--- a/Doc/library/cmath.rst
+++ b/Doc/library/cmath.rst
@@ -102,11 +102,11 @@ 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 along the negative real axis to -∞.
+   Return 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
+   along the negative real axis to -∞, continuous from above.
 
 
 .. function:: log10(x)
diff --git a/Doc/library/math.rst b/Doc/library/math.rst
index 797f32408eac3d..bd9586cefb0bbb 100644
--- a/Doc/library/math.rst
+++ b/Doc/library/math.rst
@@ -393,13 +393,10 @@ Power and logarithmic functions
    .. versionadded:: 3.2
 
 
-.. function:: log(x[, base])
-
-   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)``.
+.. function:: log(x, base=None)
 
+   Return the logarithm of *x* to the given *base*. If the *base* is not
+   specified, returns the natural logarithm of *x*.
 
 .. function:: log1p(x)
 
diff --git a/Lib/test/test_math.py b/Lib/test/test_math.py
index 433161c2dd4145..8d849045b2d11f 100644
--- a/Lib/test/test_math.py
+++ b/Lib/test/test_math.py
@@ -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)
diff --git a/Misc/NEWS.d/next/Library/2023-01-16-10-42-58.gh-issue-89381.lM2WL0.rst b/Misc/NEWS.d/next/Library/2023-01-16-10-42-58.gh-issue-89381.lM2WL0.rst
new file mode 100644
index 00000000000000..7bffe7d226e38a
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2023-01-16-10-42-58.gh-issue-89381.lM2WL0.rst
@@ -0,0 +1 @@
+:func:`~math.log` and :func:`~cmath.log` support default base=None values.
diff --git a/Modules/clinic/cmathmodule.c.h b/Modules/clinic/cmathmodule.c.h
index 941448e76e80de..4cf17c4166abf3 100644
--- a/Modules/clinic/cmathmodule.c.h
+++ b/Modules/clinic/cmathmodule.c.h
@@ -639,12 +639,10 @@ cmath_tanh(PyObject *module, PyObject *arg)
 }
 
 PyDoc_STRVAR(cmath_log__doc__,
-"log($module, z, base=<unrepresentable>, /)\n"
+"log($module, z, base=None, /)\n"
 "--\n"
 "\n"
-"log(z[, base]) -> the logarithm of z to the given base.\n"
-"\n"
-"If the base is not specified, returns the natural logarithm (base e) of z.");
+"Return the logarithm of z to the given base or the natural logarithm of z.");
 
 #define CMATH_LOG_METHODDEF    \
     {"log", _PyCFunction_CAST(cmath_log), METH_FASTCALL, cmath_log__doc__},
@@ -657,7 +655,7 @@ cmath_log(PyObject *module, PyObject *const *args, Py_ssize_t nargs)
 {
     PyObject *return_value = NULL;
     Py_complex x;
-    PyObject *y_obj = NULL;
+    PyObject *y_obj = Py_None;
 
     if (!_PyArg_CheckPositional("log", nargs, 1, 2)) {
         goto exit;
@@ -982,4 +980,4 @@ cmath_isclose(PyObject *module, PyObject *const *args, Py_ssize_t nargs, PyObjec
 exit:
     return return_value;
 }
-/*[clinic end generated code: output=87f609786ef270cd input=a9049054013a1b77]*/
+/*[clinic end generated code: output=dd34acdabdefc093 input=a9049054013a1b77]*/
diff --git a/Modules/clinic/mathmodule.c.h b/Modules/clinic/mathmodule.c.h
index 1f9725883b9820..04c5b84c30bebd 100644
--- a/Modules/clinic/mathmodule.c.h
+++ b/Modules/clinic/mathmodule.c.h
@@ -187,43 +187,34 @@ math_modf(PyObject *module, PyObject *arg)
 }
 
 PyDoc_STRVAR(math_log__doc__,
-"log(x, [base=math.e])\n"
-"Return the logarithm of x to the given base.\n"
+"log($module, x, base=None, /)\n"
+"--\n"
 "\n"
-"If the base not specified, returns the natural logarithm (base e) of x.");
+"Return the logarithm of x to the given base or the natural logarithm of x.");
 
 #define MATH_LOG_METHODDEF    \
-    {"log", (PyCFunction)math_log, METH_VARARGS, math_log__doc__},
+    {"log", _PyCFunction_CAST(math_log), METH_FASTCALL, math_log__doc__},
 
 static PyObject *
-math_log_impl(PyObject *module, PyObject *x, int group_right_1,
-              PyObject *base);
+math_log_impl(PyObject *module, PyObject *x, PyObject *base);
 
 static PyObject *
-math_log(PyObject *module, PyObject *args)
+math_log(PyObject *module, PyObject *const *args, Py_ssize_t nargs)
 {
     PyObject *return_value = NULL;
     PyObject *x;
-    int group_right_1 = 0;
-    PyObject *base = NULL;
+    PyObject *base = Py_None;
 
-    switch (PyTuple_GET_SIZE(args)) {
-        case 1:
-            if (!PyArg_ParseTuple(args, "O:log", &x)) {
-                goto exit;
-            }
-            break;
-        case 2:
-            if (!PyArg_ParseTuple(args, "OO:log", &x, &base)) {
-                goto exit;
-            }
-            group_right_1 = 1;
-            break;
-        default:
-            PyErr_SetString(PyExc_TypeError, "math.log requires 1 to 2 arguments");
-            goto exit;
+    if (!_PyArg_CheckPositional("log", nargs, 1, 2)) {
+        goto exit;
     }
-    return_value = math_log_impl(module, x, group_right_1, base);
+    x = args[0];
+    if (nargs < 2) {
+        goto skip_optional;
+    }
+    base = args[1];
+skip_optional:
+    return_value = math_log_impl(module, x, base);
 
 exit:
     return return_value;
@@ -954,4 +945,4 @@ math_ulp(PyObject *module, PyObject *arg)
 exit:
     return return_value;
 }
-/*[clinic end generated code: output=899211ec70e4506c input=a9049054013a1b77]*/
+/*[clinic end generated code: output=7c9d6093ce1c36eb input=a9049054013a1b77]*/
diff --git a/Modules/cmathmodule.c b/Modules/cmathmodule.c
index b4f7e5424b4ccf..04ecd3cb4f29ca 100644
--- a/Modules/cmathmodule.c
+++ b/Modules/cmathmodule.c
@@ -952,23 +952,21 @@ 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 is not specified, returns the natural logarithm (base e) of z.
+Return the logarithm of z to the given base or the natural logarithm 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=e1f81d4fcfd26497]*/
+/*[clinic end generated code: output=4effdb7d258e0d94 input=3b21afa2e8548074]*/
 {
     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;
diff --git a/Modules/mathmodule.c b/Modules/mathmodule.c
index 544560e8322c72..fd4d2b399e688d 100644
--- a/Modules/mathmodule.c
+++ b/Modules/mathmodule.c
@@ -2181,26 +2181,21 @@ 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.
+Return the logarithm of x to the given base or the natural logarithm 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=1ccc0718b30ca6c5]*/
 {
     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);