diff --git a/lib/matplotlib/tests/test_ticker.py b/lib/matplotlib/tests/test_ticker.py
index 222a0d7e11b0..376c03478032 100644
--- a/lib/matplotlib/tests/test_ticker.py
+++ b/lib/matplotlib/tests/test_ticker.py
@@ -1003,6 +1003,7 @@ class TestLogFormatterMathtext:
         (3, 1e2, '$\\mathdefault{100}$'),
         (3, 1e-3, '$\\mathdefault{10^{-3}}$'),
         (3, 1e3, '$\\mathdefault{10^{3}}$'),
+        (2, np.inf, ''),
     ]
 
     @pytest.mark.parametrize('min_exponent, value, expected', test_data)
@@ -1259,6 +1260,13 @@ def test_LogFormatter_call_tiny(self, val):
         temp_lf.axis.set_view_interval(1, 10)
         temp_lf(val)
 
+    @pytest.mark.parametrize('val', [1e+323, 2e+323, 10e+323, 11e+323])
+    def test_LogFormatter_call_big(self, val):
+        # test coeff computation in __call__
+        temp_lf = mticker.LogFormatter()
+        temp_lf.create_dummy_axis()
+        temp_lf(val)
+
 
 class TestLogitFormatter:
     @staticmethod
@@ -1835,6 +1843,27 @@ def test_small_range_loglocator(numticks):
         assert (np.diff(np.log10(ll.tick_values(6, 150))) == 1).all()
 
 
+# https://github.com/matplotlib/matplotlib/pull/27609
+# TODO: This test currently fails, as expected and it needs to be fixed...
+# To do this completely correctly, we should figure out what are our limits
+# for when do integers are interpreted eventually as infinities and why,
+# and perhaps always work with np.float128 to increase accuracy as much as
+# possible. Eitherway, we should document what is our limit and how it is
+# related to the accuracy of np.float{64,128} and Python's native float
+# which seems to be the same as np.float64 for this purpose...
+def test_LogFormatter_almost_inf():
+    fig, ax = plt.subplots()
+    # TODO: Figure out why 1e400 won't fail, but will make the ax.plot describe
+    # 1e400 as inf (and hence not print it, and hence not fail).
+    ax.plot([1, 2], [1, 1e300])
+    ax.set_yscale("log")
+    fig.draw_without_rendering()
+    almost_inf = ax.get_lines()[0].get_ydata()[1]
+    # TODO/WIP: remove of course when the above TODOs are fixed. Perhaps assert
+    # something with ydata...
+    assert True == False
+
+
 def test_NullFormatter():
     formatter = mticker.NullFormatter()
     assert formatter(1.0) == ''
diff --git a/lib/matplotlib/ticker.py b/lib/matplotlib/ticker.py
index 0053031ece3e..63a519f7933f 100644
--- a/lib/matplotlib/ticker.py
+++ b/lib/matplotlib/ticker.py
@@ -1001,6 +1001,8 @@ def __call__(self, x, pos=None):
             return '0'
 
         x = abs(x)
+        if not np.isfinite(x):
+            return ''
         b = self._base
         # only label the decades
         fx = math.log(x) / math.log(b)
@@ -1081,6 +1083,8 @@ def __call__(self, x, pos=None):
 
         sign_string = '-' if x < 0 else ''
         x = abs(x)
+        if not np.isfinite(x):
+            return ''
         b = self._base
 
         # only label the decades