diff --git a/lib/matplotlib/tests/test_transforms.py b/lib/matplotlib/tests/test_transforms.py
index af9cea700a38..c138b5937db9 100644
--- a/lib/matplotlib/tests/test_transforms.py
+++ b/lib/matplotlib/tests/test_transforms.py
@@ -492,6 +492,16 @@ def test_transform_single_point():
     r = t.transform_affine((1, 1))
     assert r.shape == (2,)
 
+
+@cleanup
+def test_log_transform():
+    # Tests that the last line runs without exception (previously the
+    # transform would fail if one of the axes was logarithmic).
+    fig, ax = plt.subplots()
+    ax.set_yscale('log')
+    ax.transData.transform((1,1))
+
+
 if __name__=='__main__':
     import nose
     nose.runmodule(argv=['-s','--with-doctest'], exit=False)
diff --git a/lib/matplotlib/transforms.py b/lib/matplotlib/transforms.py
index c8848eca900f..0c7a48cbf7c7 100644
--- a/lib/matplotlib/transforms.py
+++ b/lib/matplotlib/transforms.py
@@ -1285,8 +1285,33 @@ def transform(self, values):
 
         Accepts a numpy array of shape (N x :attr:`input_dims`) and
         returns a numpy array of shape (N x :attr:`output_dims`).
-        """
-        return self.transform_affine(self.transform_non_affine(values))
+
+        Alternatively, accepts a numpy array of length :attr:`input_dims`
+        and returns a numpy array of length :attr:`output_dims`.
+        """
+        # Ensure that values is a 2d array (but remember whether
+        # we started with a 1d or 2d array).
+        values = np.asanyarray(values)
+        ndim = values.ndim
+        values = values.reshape((-1, self.input_dims))
+
+        # Transform the values
+        res = self.transform_affine(self.transform_non_affine(values))
+
+        # Convert the result back to the shape of the input values.
+        if ndim == 0:
+            assert not np.ma.is_masked(res)  # just to be on the safe side                                             
+            return res[0, 0]
+        if ndim == 1:
+            return res.reshape(-1)
+        elif ndim == 2:
+            return res
+        else:
+            raise ValueError(
+                "Input values must have shape (N x {dims}) "
+                "or ({dims}).".format(dims=self.input_dims))
+
+        return res
 
     def transform_affine(self, values):
         """
@@ -1302,6 +1327,9 @@ def transform_affine(self, values):
 
         Accepts a numpy array of shape (N x :attr:`input_dims`) and
         returns a numpy array of shape (N x :attr:`output_dims`).
+
+        Alternatively, accepts a numpy array of length :attr:`input_dims`
+        and returns a numpy array of length :attr:`output_dims`.
         """
         return self.get_affine().transform(values)
 
@@ -1318,6 +1346,9 @@ def transform_non_affine(self, values):
 
         Accepts a numpy array of shape (N x :attr:`input_dims`) and
         returns a numpy array of shape (N x :attr:`output_dims`).
+
+        Alternatively, accepts a numpy array of length :attr:`input_dims`
+        and returns a numpy array of length :attr:`output_dims`.
         """
         return values
 
@@ -1944,7 +1975,7 @@ def get_matrix(self):
     get_matrix.__doc__ = Affine2DBase.get_matrix.__doc__
 
     def transform(self, points):
-        return points
+        return np.asanyarray(points)
     transform.__doc__ = Affine2DBase.transform.__doc__
 
     transform_affine = transform