diff --git a/lib/matplotlib/scale.py b/lib/matplotlib/scale.py
index d86de461efc8..f3960e359bb7 100644
--- a/lib/matplotlib/scale.py
+++ b/lib/matplotlib/scale.py
@@ -288,7 +288,7 @@ def __init__(self, axis, *, base=10, subs=None, nonpositive="clip"):
     def set_default_locators_and_formatters(self, axis):
         # docstring inherited
         axis.set_major_locator(LogLocator(self.base))
-        axis.set_major_formatter(LogFormatterSciNotation(self.base))
+        axis.set_major_formatter(LogFormatterSciNotation(self.base, labelOnlyBase=True))
         axis.set_minor_locator(LogLocator(self.base, self.subs))
         axis.set_minor_formatter(
             LogFormatterSciNotation(self.base,
@@ -451,7 +451,7 @@ def __init__(self, axis, *, base=10, linthresh=2, subs=None, linscale=1):
     def set_default_locators_and_formatters(self, axis):
         # docstring inherited
         axis.set_major_locator(SymmetricalLogLocator(self.get_transform()))
-        axis.set_major_formatter(LogFormatterSciNotation(self.base))
+        axis.set_major_formatter(LogFormatterSciNotation(self.base, labelOnlyBase=True))
         axis.set_minor_locator(SymmetricalLogLocator(self.get_transform(),
                                                      self.subs))
         axis.set_minor_formatter(NullFormatter())
diff --git a/lib/matplotlib/tests/test_ticker.py b/lib/matplotlib/tests/test_ticker.py
index 9a1c8f0f5881..64440c899e3d 100644
--- a/lib/matplotlib/tests/test_ticker.py
+++ b/lib/matplotlib/tests/test_ticker.py
@@ -1155,6 +1155,13 @@ def _sub_labels(self, axis, subs=()):
         label_test = [fmt(x) != '' for x in minor_tlocs]
         assert label_test == label_expected
 
+    def _no_minor_labels(self, axis):
+        fmt = axis.get_minor_formatter()
+        minor_tlocs = axis.get_minorticklocs()
+        fmt.set_locs(minor_tlocs)
+        label_test = [fmt(x) == '' for x in minor_tlocs]
+        assert all(label_test)
+
     @mpl.style.context('default')
     def test_sublabel(self):
         # test label locator
@@ -1193,6 +1200,15 @@ def test_sublabel(self):
         ax.set_xlim(0.5, 0.9)
         self._sub_labels(ax.xaxis, subs=np.arange(2, 10, dtype=int))
 
+        # minor_thresholds=(0, 0), no minor tick will be labeled
+        ax.xaxis.set_minor_formatter(
+            mticker.LogFormatter(
+                labelOnlyBase=False,
+                minor_thresholds=(0, 0)
+            )
+        )
+        self._no_minor_labels(ax.xaxis)
+
     @pytest.mark.parametrize('val', [1, 10, 100, 1000])
     def test_LogFormatter_call(self, val):
         # test _num_to_string method used in __call__
diff --git a/lib/matplotlib/ticker.py b/lib/matplotlib/ticker.py
index b4793929de9b..6d197d732d50 100644
--- a/lib/matplotlib/ticker.py
+++ b/lib/matplotlib/ticker.py
@@ -952,8 +952,8 @@ def set_locs(self, locs=None):
             numdec = abs(vmax - vmin)
 
         if numdec > self.minor_thresholds[0]:
-            # Label only bases
-            self._sublabels = {1}
+            # No minor ticks will be labeled
+            self._sublabels = {}
         elif numdec > self.minor_thresholds[1]:
             # Add labels between bases at log-spaced coefficients;
             # include base powers in case the locations include
@@ -987,9 +987,10 @@ def __call__(self, x, pos=None):
         exponent = round(fx) if is_x_decade else np.floor(fx)
         coeff = round(b ** (fx - exponent))
 
-        if self.labelOnlyBase and not is_x_decade:
-            return ''
-        if self._sublabels is not None and coeff not in self._sublabels:
+        if self.labelOnlyBase:
+            if not is_x_decade:
+                return ''
+        elif self._sublabels is not None and coeff not in self._sublabels:
             return ''
 
         vmin, vmax = self.axis.get_view_interval()
@@ -1073,9 +1074,10 @@ def __call__(self, x, pos=None):
         if is_x_decade:
             fx = round(fx)
 
-        if self.labelOnlyBase and not is_x_decade:
-            return ''
-        if self._sublabels is not None and coeff not in self._sublabels:
+        if self.labelOnlyBase:
+            if not is_x_decade:
+                return ''
+        elif self._sublabels is not None and coeff not in self._sublabels:
             return ''
 
         # use string formatting of the base if it is not an integer