[ENH]: Compute ticks of log scaled axes a bit better? #29414

Problem

Taken from here, I found the following ticks (from plt.semilogy([1.5, 50])):

Not informative enough. I wish there was an easy way to set_minor_formatter such that say e.g 2x10^0, 3x10^0 would have been displayed. I tried to use (full MWE this time):

import matplotlib.pyplot as plt
from matplotlib import ticker

plt.semilogy([1.5, 50])
plt.gca().yaxis.set_minor_formatter(ticker.LogFormatter(base=10, labelOnlyBase=True))
plt.show()

But it changed nothing. OTH I was capable to obtain my desired output with:

#!/usr/bin/env python

import numpy as np

import matplotlib.pyplot as plt
from matplotlib import ticker

plt.semilogy([1.5, 50])
def get_minor_tick_string(x, pos):
    b = np.floor(np.log10(x)).astype(int)
    p = pos % 8
    if p < 3:
        return f"${p+2}\\cdot 10^{b}$"
    else:
        return ""
plt.gca().yaxis.set_minor_formatter(get_minor_tick_string)
plt.show()

But I'm sure it is not very versatile. It produces:

Which is not bad.

Proposed solution

I'd like to first explain in the docs / fix the behavior of set_minor_formatter(ticker.LogFormatter(base=10, labelOnlyBase=True)). 2ndly, I'd like to discuss the possibility of adding tick labels as shown above, because I think I'm not the only one that was not satisfied by this default behavior.