Change of basis for log-log plotting, grid in log-scale

I want to plot in log scale in both axis, when using the default base, the graph plotted is correct, but since the y variation of my data is small I would like to change for base=2, when I does this the grid is no longer in log scale but evenly spaced with no small ticks (which makes easier to see that this is a log scale plot)

where is my code for default base

import numpy as np
import matplotlib.pyplot as plt

x = np.array([8, 16, 32, 64, 128, 256, 512, 1024])
y = np.array([2.412,  3.424,  5.074,  7.308, 11.444, 18.394, 30.644, 48.908])
base = 2 # Base for logarithmic scale

fig, ax = plt.subplots(figsize=(8,6))

# log x and y axis
ax.plot(x, y, 'o--')
ax.set_xscale('log')
ax.set_yscale('log')
ax.set(title='loglog')
ax.grid()
ax.grid(which="minor", color="0.9")

when I change the base, by using the argument base in ax.set_xscale()

import numpy as np
import matplotlib.pyplot as plt

x = np.array([8, 16, 32, 64, 128, 256, 512, 1024])
y = np.array([2.412,  3.424,  5.074,  7.308, 11.444, 18.394, 30.644, 48.908])
base = 2 # Base for logarithmic scale

fig, ax = plt.subplots(figsize=(8,6))

# log x and y axis
ax.plot(x, y, 'o--')
ax.set_xscale('log', base=base)
ax.set_yscale('log', base=base)
ax.set(title='loglog')
ax.grid()
ax.grid(which="minor", color="0.9")

The graphs I created are correct, what I want is aesthetic. Keep these little ticks "logarithmly" spaced.
I have tried to use ax.loglog(x, y, '0--', base=base) instead of ax.plot(x, y, 'o--'), and also tried to manually set the tickers with

def ticks(y, pos):
    return r'$2^{:.0f}$'.format(np.log(y)/np.log(base)) if y > 0 else '0'

ax.xaxis.set_major_formatter(ticker.FuncFormatter(ticks))
ax.yaxis.set_major_formatter(ticker.FuncFormatter(ticks))

None of that work.

What I want is aesthetics, both graphs are correct, but these little ticks "logarithmly" spaced makes the graph easier to interpret as log-log plot.

As @JohanC answered with base=2 there are no logical positions to place these minor ticks, so my solution was to use `basex=8` for `x` axis and `basey=4` for `y` axis.

basex = 8 # Base for logarithmic scale

basey = 4 # Base for logarithmic scale

fig, ax = plt.subplots(figsize=(8,6))

ax.plot(x, y, 'o--')

ax.set_xscale('log',base = basex)

ax.set_yscale('log', base = basey)

ax.set(title='loglog')

ax.grid()

ax.grid(which="minor", color="0.9")