diff --git a/galleries/examples/scales/log_demo.py b/galleries/examples/scales/log_demo.py
@@ -1,47 +1,83 @@
 """
-========
-Log Demo
-========
+=========
+Log scale
+=========
 
 Examples of plots with logarithmic axes.
+
+You can set the x/y axes to be logarithmic by passing "log" to `~.Axes.set_xscale` /
+`~.Axes.set_yscale`.
+
+Convenience functions ``semilogx``, ``semilogy``, and ``loglog``
+----------------------------------------------------------------
+Since plotting data on semi-logarithmic or double-logarithmic scales is very common,
+the functions `~.Axes.semilogx`, `~.Axes.semilogy`, and `~.Axes.loglog` are shortcuts
+for setting the scale and plotting data; e.g. ``ax.semilogx(x, y)`` is equivalent to
+``ax.set_xscale('log'); ax.plot(x, y)``.
 """
 
 import matplotlib.pyplot as plt
 import numpy as np
 
-# Data for plotting
-t = np.arange(0.01, 20.0, 0.01)
-
-# Create figure
-fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2, 2)
-
-# log y axis
-ax1.semilogy(t, np.exp(-t / 5.0))
-ax1.set(title='semilogy')
+fig, (ax1, ax2, ax3) = plt.subplots(1, 3, layout='constrained', figsize=(7, 7/3))
+# log x axis
+t = np.arange(0.01, 10.0, 0.01)
+ax1.semilogx(t, np.sin(2 * np.pi * t))
+ax1.set(title='semilogx')
 ax1.grid()
+ax1.grid(which="minor", color="0.9")
 
-# log x axis
-ax2.semilogx(t, np.sin(2 * np.pi * t))
-ax2.set(title='semilogx')
+# log y axis
+x = np.arange(4)
+ax2.semilogy(4*x, 10**x, 'o--')
+ax2.set(title='semilogy')
 ax2.grid()
+ax2.grid(which="minor", color="0.9")
 
 # log x and y axis
-ax3.loglog(t, 20 * np.exp(-t / 10.0))
-ax3.set_xscale('log', base=2)
-ax3.set(title='loglog base 2 on x')
+x = np.array([1, 10, 100, 1000])
+ax3.loglog(x, 5 * x, 'o--')
+ax3.set(title='loglog')
 ax3.grid()
+ax3.grid(which="minor", color="0.9")
+
+# %%
+# Logarithms with other bases
+# ---------------------------
+# By default, the log scale is to the base 10. One can change this via the *base*
+# parameter.
+fig, ax = plt.subplots()
+ax.bar(["L1 cache", "L2 cache", "L3 cache", "RAM", "SSD"],
+       [32, 1_000, 32_000, 16_000_000, 512_000_000])
+ax.set_yscale('log', base=2)
+ax.set_yticks([1, 2**10, 2**20, 2**30], labels=['kB', 'MB', 'GB', 'TB'])
+ax.set_title("Typical memory sizes")
+ax.yaxis.grid()
+
+# %%
+# Dealing with negative values
+# ----------------------------
+# Non-positive values cannot be displayed on a log scale. The scale has two options
+# to handle these. Either mask the values so that they are ignored, or clip them
+# to a small positive value. Which one is more suited depends on the type of the
+# data and the visualization.
+#
+# The following example contains errorbars going negative. If we mask these values,
+# the bar vanishes, which is not desirable. In contrast, clipping makes the value
+# small positive (but well below the used scale) so that the error bar is drawn
+# to the edge of the Axes.
+x = np.linspace(0.0, 2.0, 10)
+y = 10**x
+yerr = 1.75 + 0.75*y
 
-# With errorbars: clip non-positive values
-# Use new data for plotting
-x = 10.0**np.linspace(0.0, 2.0, 20)
-y = x**2.0
+fig, (ax1, ax2) = plt.subplots(1, 2, layout="constrained", figsize=(6, 3))
+fig.suptitle("errorbars going negative")
+ax1.set_yscale("log", nonpositive='mask')
+ax1.set_title('nonpositive="mask"')
+ax1.errorbar(x, y, yerr=yerr, fmt='o', capsize=5)
 
-ax4.set_xscale("log", nonpositive='clip')
-ax4.set_yscale("log", nonpositive='clip')
-ax4.set(title='Errorbars go negative')
-ax4.errorbar(x, y, xerr=0.1 * x, yerr=5.0 + 0.75 * y)
-# ylim must be set after errorbar to allow errorbar to autoscale limits
-ax4.set_ylim(bottom=0.1)
+ax2.set_yscale("log", nonpositive='clip')
+ax2.set_title('nonpositive="clip"')
+ax2.errorbar(x, y, yerr=yerr, fmt='o', capsize=5)
 
-fig.tight_layout()
 plt.show()
diff --git a/galleries/examples/scales/semilogx_demo.py b/galleries/examples/scales/semilogx_demo.py