Custom ticks label in scientific notation while using log scale

I'm having trouble with matplotlib (version 3.1.3) : I would like to add custom ticks and tick labels on a log scale axis while preserving scientific notation.

To say it otherwise: I want to add custom ticks on a log scale axis and label them using the good old-fashioned '%1.1e' (or whatever numbers) format but, for instance, instead of having '2.5e-02', I would like to have '2.5 x 10^-2' (or '2.5 \times 10^{-2}' in latex).

So I start with a minimum working piece of code without custom ticks:

import matplotlib as mpl
import matplotlib.pyplot as plt
print('MATPLOTLIB  VERSION : %s' % mpl.__version__)

plt.style.use("default")

# DATA
x = [0.1, 0.075, 0.05, 0.025, 0.01, 0.0075, 0.005, 0.0025, 0.001, 0.00075, 0.0005, 0.00025, 0.0001, 7.5e-05, 5e-05, 2.5e-05, 1e-05, 1e-06, 1e-07, 1e-08, 1e-09, 1e-10]
y = x

fig = plt.figure()
ax = plt.axes()
plt.loglog()
plt.minorticks_off()
path = ax.plot(x, y)

plt.savefig('test.png')

which gives:

Nice but, as I said, I would like to add custom ticks on the xaxis. More precisely, I would like to put limits to the axis and define equally log-spaced labels between those limits. Let's say I want 4 ticks; it gives the following piece of code:

import matplotlib as mpl
import matplotlib.pyplot as plt
print('MATPLOTLIB  VERSION : %s' % mpl.__version__)

plt.style.use("default")

# DATA
x = [0.1, 0.075, 0.05, 0.025, 0.01, 0.0075, 0.005, 0.0025, 0.001, 0.00075, 0.0005, 0.00025, 0.0001, 7.5e-05, 5e-05, 2.5e-05, 1e-05, 1e-06, 1e-07, 1e-08, 1e-09, 1e-10]
y = x

xmin = min(x)
xmax = max(x)
ymin = min(y)
ymax = max(y)

# XTICKS
nbdiv = 4
xTicks = []
k = pow((xmin/xmax),1./(nbdiv-1.))
for i in range(0,nbdiv):
  xTicks.append(xmax*pow(k,i))

# PLOT
fig = plt.figure()
ax = plt.axes()
plt.loglog()
plt.minorticks_off()
plt.axis([xmin,xmax,ymin,ymax])
plt.xticks(xTicks)
path = ax.plot(x, y)

plt.savefig('test_working_4.png')

which provides the following plot:

That's kind of the result I wanted to obtain. However, if the number of ticks (nbdiv) changes, for instance becomes equal to 5, I get:

And this time only the first and last ticks are labelled. It appears that only numbers which are equal (or at least close) to an integer power of ten (10^n) are labelled. I tried to change the default tick format with a matplot.ticker.ScalarFormatter but I didn't manage to tune it to solve this problem. I also tried LogFormatterMathText and LogFormatterSciNotation, it wasn't better.

The issue in itself doesn't seem so difficult to me, so I don't understand why I'm having so much trouble... What am I doing wrong? Does someone has keys to make me understand my errors?

In any case, I thank for reading and I thank you in advance for your response.

P.S.: Sorry for potential english mistakes, it's not my native language.

Solved, based on your code above. This one is much simpler. You need to use xticklabels.

%matplotlib inline
import matplotlib as mpl
import matplotlib.pyplot as plt
from sympy import pretty_print as pp, latex
print('MATPLOTLIB  VERSION : %s' % mpl.__version__)

plt.style.use("default")

# DATA
x = [0.1, 0.075, 0.05, 0.025, 0.01, 0.0075, 0.005, 0.0025, 0.001, 0.00075, 0.0005, 0.00025, 0.0001, 7.5e-05, 5e-05, 2.5e-05, 1e-05, 1e-06, 1e-07, 1e-08, 1e-09, 1e-10]
y = x

xmin = min(x)
xmax = max(x)
ymin = min(y)
ymax = max(y)

# XTICKS
nbdiv = 5
xTicks = []
xticklabels = []
k = pow((xmin/xmax),1./(nbdiv-1.))
for i in range(0,nbdiv):
  xTicks.append(xmax*pow(k,i))
  printstr = '{:.2e}'.format(xmax*pow(k,i))
  ls = printstr.split('e')
  xticklabels.append((ls[0]+' x $10^{'  +ls[1] + '}$'))

# PLOT
fig = plt.figure()
ax = plt.axes()
plt.loglog()
plt.minorticks_off()
plt.axis([xmin,xmax,ymin,ymax])
plt.xticks(xTicks)
path = ax.plot(x, y)

plt.savefig('test_working_4.png')
ax.set_xticklabels(xticklabels)