diff --git a/galleries/examples/scales/custom_scale.py b/galleries/examples/scales/custom_scale.py
@@ -1,4 +1,6 @@
 """
+.. _custom_scale:
+
 ============
 Custom scale
 ============

diff --git a/galleries/users_explain/artists/artist_intro.rst b/galleries/users_explain/artists/artist_intro.rst
@@ -5,9 +5,8 @@ Introduction to Artists
 
 Almost all objects you interact with on a Matplotlib plot are called "Artist"
 (and are subclasses of the `.Artist` class).  :doc:`Figure <../figure/index>`
-and :doc:`Axes <../axes/index>` are Artists, and generally contain :doc:`Axis
-<../axis/index>` Artists and Artists that contain data or annotation
-information.
+and :doc:`Axes <../axes/index>` are Artists, and generally contain
+`~.axis.Axis` Artists and Artists that contain data or annotation information.
 
 
 Creating Artists

diff --git a/galleries/users_explain/artists/index.rst b/galleries/users_explain/artists/index.rst
@@ -4,9 +4,8 @@ Artists
 
 Almost all objects you interact with on a Matplotlib plot are called "Artist"
 (and are subclasses of the `.Artist` class).  :doc:`Figure <../figure/index>`
-and :doc:`Axes <../axes/index>` are Artists, and generally contain :doc:`Axis
-<../axis/index>` Artists and Artists that contain data or annotation
-information.
+and :doc:`Axes <../axes/index>` are Artists, and generally contain
+`~.axis.Axis` Artists and Artists that contain data or annotation information.
 
 .. toctree::
     :maxdepth: 2

diff --git a/galleries/users_explain/axes/axes_intro.rst b/galleries/users_explain/axes/axes_intro.rst
@@ -5,7 +5,7 @@ Introduction to Axes (or Subplots)
 
 Matplotlib `~.axes.Axes` are the gateway to creating your data visualizations.
 Once an Axes is placed on a figure there are many methods that can be used to
-add data to the Axes. An Axes typically has a pair of :doc:`Axis <../axis/index>`
+add data to the Axes. An Axes typically has a pair of `~.axis.Axis`
 Artists that define the data coordinate system, and include methods to add
 annotations like x- and y-labels, titles, and legends.
 
@@ -130,7 +130,7 @@ Note that text can also be added to axes using `~.axes.Axes.text`, and `~.axes.A
 Axes limits, scales, and ticking
 --------------------------------
 
-Each Axes has two (or more) `~.axis.Axis` objects, that can be accessed via :attr:`~matplotlib.axes.Axes.xaxis` and :attr:`~matplotlib.axes.Axes.yaxis` properties.  These have substantial number of methods on them, and for highly customizable Axis-es it is useful to read more about that API (:doc:`../axis/index`).   However, the Axes class offers a number of helpers for the most common of these methods.  Indeed, the `~.axes.Axes.set_xlabel`, discussed above, is a helper for the `~.Axis.set_label_text`.
+Each Axes has two (or more) `~.axis.Axis` objects, that can be accessed via :attr:`~matplotlib.axes.Axes.xaxis` and :attr:`~matplotlib.axes.Axes.yaxis` properties.  These have substantial number of methods on them, and for highly customizable Axis-es it is useful to read the API at `~.axis.Axis`.   However, the Axes class offers a number of helpers for the most common of these methods.  Indeed, the `~.axes.Axes.set_xlabel`, discussed above, is a helper for the `~.Axis.set_label_text`.
 
 Other important methods set the extent on the axes (`~.axes.Axes.set_xlim`, `~.axes.Axes.set_ylim`), or more fundamentally the scale of the axes.  So for instance, we can make an Axis have a logarithmic scale, and zoom in on a sub-portion of the data:
 
@@ -158,7 +158,7 @@ Many aspects of Axes ticks and tick labeling can be adjusted using `~.axes.Axes.
                    labelcolor='green')
 
 
-More fine-grained control on ticks, setting scales, and controlling the Axis can be highly customized beyond these Axes-level helpers.  An introduction to these methods can be found in :ref:`users_axis`, or the API reference for `.axis.Axis`.
+More fine-grained control on ticks, setting scales, and controlling the Axis can be highly customized beyond these Axes-level helpers.
 
 Axes layout
 -----------

diff --git a/galleries/users_explain/axes/axes_scales.py b/galleries/users_explain/axes/axes_scales.py
@@ -0,0 +1,223 @@
+"""
+.. _user_axes_scales:
+
+===========
+Axis scales
+===========
+
+By default Matplotlib displays data on the axis using a linear scale.
+Matplotlib also supports `logarithmic scales
+<https://en.wikipedia.org/wiki/Logarithmic_scale>`_, and other less common
+scales as well. Usually this can be done directly by using the
+`~.axes.Axes.set_xscale` or `~.axes.Axes.set_yscale` methods.
+
+"""
+import matplotlib.pyplot as plt
+import numpy as np
+
+import matplotlib.scale as mscale
+from matplotlib.ticker import FixedLocator, NullFormatter
+
+fig, axs = plt.subplot_mosaic([['linear', 'linear-log'],
+                               ['log-linear', 'log-log']], layout='constrained')
+
+x = np.arange(0, 3*np.pi, 0.1)
+y = 2 * np.sin(x) + 3
+
+ax = axs['linear']
+ax.plot(x, y)
+ax.set_xlabel('linear')
+ax.set_ylabel('linear')
+
+ax = axs['linear-log']
+ax.plot(x, y)
+ax.set_yscale('log')
+ax.set_xlabel('linear')
+ax.set_ylabel('log')
+
+ax = axs['log-linear']
+ax.plot(x, y)
+ax.set_xscale('log')
+ax.set_xlabel('log')
+ax.set_ylabel('linear')
+
+ax = axs['log-log']
+ax.plot(x, y)
+ax.set_xscale('log')
+ax.set_yscale('log')
+ax.set_xlabel('log')
+ax.set_ylabel('log')
+
+# %%
+# loglog and semilogx/y
+# =====================
+#
+# The logarithmic axis is used so often that there are a set
+# helper functions, that do the same thing: `~.axes.Axes.semilogy`,
+# `~.axes.Axes.semilogx`, and `~.axes.Axes.loglog`.
+
+fig, axs = plt.subplot_mosaic([['linear', 'linear-log'],
+                               ['log-linear', 'log-log']], layout='constrained')
+
+x = np.arange(0, 3*np.pi, 0.1)
+y = 2 * np.sin(x) + 3
+
+ax = axs['linear']
+ax.plot(x, y)
+ax.set_xlabel('linear')
+ax.set_ylabel('linear')
+ax.set_title('plot(x, y)')
+
+ax = axs['linear-log']
+ax.semilogy(x, y)
+ax.set_xlabel('linear')
+ax.set_ylabel('log')
+ax.set_title('semilogy(x, y)')
+
+ax = axs['log-linear']
+ax.semilogx(x, y)
+ax.set_xlabel('log')
+ax.set_ylabel('linear')
+ax.set_title('semilogx(x, y)')
+
+ax = axs['log-log']
+ax.loglog(x, y)
+ax.set_xlabel('log')
+ax.set_ylabel('log')
+ax.set_title('loglog(x, y)')
+
+# %%
+# Other built-in scales
+# =====================
+#
+# There are other scales that can be used.  The list of registered
+# scales can be returned from `.scale.get_scale_names`:
+
+print(mscale.get_scale_names())
+
+# %%
+#
+
+todo = ['asinh', 'symlog', 'log', 'logit', ]
+fig, axs = plt.subplot_mosaic([['asinh', 'symlog'],
+                               ['log', 'logit']], layout='constrained')
+
+x = np.arange(0, 1000)
+
+for td in todo:
+    ax = axs[td]
+    if td in ['asinh', 'symlog']:
+        yy = x - np.mean(x)
+    elif td in ['logit']:
+        yy = (x-np.min(x))
+        yy = yy / np.max(np.abs(yy))
+    else:
+        yy = x
+
+    ax.plot(yy, yy)
+    ax.set_yscale(td)
+    ax.set_title(td)
+
+# %%
+# Optional arguments for scales
+# =============================
+#
+# Some of the default scales have optional arguments.  These are
+# documented in the API reference for the respective scales at
+# `~.matplotlib.scale`.  One can change the base of the logarithm
+# being plotted (eg 2 below) or the linear threshold range
+# for ``'symlog'``.
+
+fig, axs = plt.subplot_mosaic([['log', 'symlog']], layout='constrained',
+                              figsize=(6.4, 3))
+
+for td in axs:
+    ax = axs[td]
+    if td in ['log']:
+        ax.plot(x, x)
+        ax.set_yscale('log', base=2)
+        ax.set_title('log base=2')
+    else:
+        ax.plot(x - np.mean(x), x - np.mean(x))
+        ax.set_yscale('symlog', linthresh=100)
+        ax.set_title('symlog linthresh=100')
+
+
+# %%
+#
+# Arbitrary function scales
+# ============================
+#
+# Users can define a full scale class and pass that to `~.axes.Axes.set_xscale`
+# and `~.axes.Axes.set_yscale` (see :ref:`custom_scale`).  A short cut for this
+# is to use the 'function' scale, and pass as extra arguments a ``forward`` and
+# an ``inverse`` function.  The following performs a `Mercator transform
+# <https://en.wikipedia.org/wiki/Mercator_projection>`_ to the y-axis.
+
+# Function Mercator transform
+def forward(a):
+    a = np.deg2rad(a)
+    return np.rad2deg(np.log(np.abs(np.tan(a) + 1.0 / np.cos(a))))
+
+
+def inverse(a):
+    a = np.deg2rad(a)
+    return np.rad2deg(np.arctan(np.sinh(a)))
+
+
+t = np.arange(0, 170.0, 0.1)
+s = t / 2.
+
+fig, ax = plt.subplots(layout='constrained')
+ax.plot(t, s, '-', lw=2)
+
+ax.set_yscale('function', functions=(forward, inverse))
+ax.set_title('function: Mercator')
+ax.grid(True)
+ax.set_xlim([0, 180])
+ax.yaxis.set_minor_formatter(NullFormatter())
+ax.yaxis.set_major_locator(FixedLocator(np.arange(0, 90, 10)))
+
+
+# %%
+#
+# What is a "scale"?
+# ==================
+#
+# A scale is an object that gets attached to an axis.  The class documentation
+# is at `~matplotlib.scale`. `~.axes.Axes.set_xscale` and `~.axes.Axes.set_yscale`
+# set the scale on the respective Axis objects.  You can determine the scale
+# on an axis with `~.axis.Axis.get_scale`:
+
+fig, ax = plt.subplots(layout='constrained',
+                              figsize=(3.2, 3))
+ax.semilogy(x, x)
+
+print(ax.xaxis.get_scale())
+print(ax.yaxis.get_scale())
+
+# %%
+#
+# Setting a scale does three things.  First it defines a transform on the axis
+# that maps between data values to position along the axis.  This transform can
+# be accessed via ``get_transform``:
+
+print(ax.yaxis.get_transform())
+
+# %%
+#
+# Transforms on the axis are a relatively low-level concept, but is one of the
+# important roles played by ``set_scale``.
+#
+# Setting the scale also sets default tick locators (`~.ticker`) and tick
+# formatters appropriate for the scale.   An axis with a 'log' scale has a
+# `~.ticker.LogLocator` to pick ticks at decade intervals, and a
+# `~.ticker.LogFormatter` to use scientific notation on the decades.
+
+print('X axis')
+print(ax.xaxis.get_major_locator())
+print(ax.xaxis.get_major_formatter())
+
+print('Y axis')
+print(ax.yaxis.get_major_locator())
+print(ax.yaxis.get_major_formatter())