set ticks with logarithmic scale

from matplotlib.ticker import ScalarFormatter, NullFormatter
for axis in [ax.xaxis]:
    axis.set_major_formatter(ScalarFormatter())
    axis.set_minor_formatter(NullFormatter())

This removes the exponential notation

import matplotlib
from matplotlib import pyplot as plt
fig1, ax1 = plt.subplots()
ax1.plot([10, 100, 1000], [1,2,3])
ax1.set_xscale('log')
ax1.set_xticks([20, 200, 500])
ax1.get_xaxis().set_major_formatter(matplotlib.ticker.ScalarFormatter())

or

ax1.get_xaxis().get_major_formatter().labelOnlyBase = False
plt.show()

I'm going to add a few plots and show how to remove the minor ticks:

The OP:

from matplotlib import pyplot as plt

fig1, ax1 = plt.subplots()
ax1.plot([10, 100, 1000], [1,2,3])
ax1.set_xscale('log')
ax1.set_xticks([20, 300, 500])
plt.show()

To add some specific ticks, as tcaswell pointed out, you can use matplotlib.ticker.ScalarFormatter:

from matplotlib import pyplot as plt
import matplotlib.ticker

fig1, ax1 = plt.subplots()
ax1.plot([10, 100, 1000], [1,2,3])
ax1.set_xscale('log')
ax1.set_xticks([20, 300, 500])
ax1.get_xaxis().set_major_formatter(matplotlib.ticker.ScalarFormatter())
plt.show()

To remove the minor ticks, you can use matplotlib.rcParams['xtick.minor.size']:

from matplotlib import pyplot as plt
import matplotlib.ticker

matplotlib.rcParams['xtick.minor.size'] = 0
matplotlib.rcParams['xtick.minor.width'] = 0

fig1, ax1 = plt.subplots()
ax1.plot([10, 100, 1000], [1,2,3])
ax1.set_xscale('log')
ax1.set_xticks([20, 300, 500])
ax1.get_xaxis().set_major_formatter(matplotlib.ticker.ScalarFormatter())

plt.show()

You could use instead ax1.get_xaxis().set_tick_params, it has the same effect (but only modifies the current axis, not all future figures unlike matplotlib.rcParams):

from matplotlib import pyplot as plt
import matplotlib.ticker

fig1, ax1 = plt.subplots()
ax1.plot([10, 100, 1000], [1,2,3])
ax1.set_xscale('log')
ax1.set_xticks([20, 300, 500])
ax1.get_xaxis().set_major_formatter(matplotlib.ticker.ScalarFormatter())

ax1.get_xaxis().set_tick_params(which='minor', size=0)
ax1.get_xaxis().set_tick_params(which='minor', width=0) 

plt.show()

Would be better to use np.geomspace as xticks

ax = sns.histplot(arr, log_scale=True)
ax.xaxis.set_major_formatter(matplotlib.ticker.ScalarFormatter())
ax.set_xticks( np.geomspace(1, 1500 ,15).round() )