Concentric layered sphere with logarithmically scaled radius

Question

How can I achieve a 3D visual of a concentric layered sphere (like an onion), where the radius of each layer is tick-marked using a logarithmic scale?

Specifically, my project is to realize the following:

1) Layer the first sphere like the concentric rings of an onion, with the innermost layer having a thickness of "e" (Euler's number), and the outermost layer at an infinite distance having a thickness of 1. So, cutoff the concentric sphere display at some "x" radius, where "x" is 1 < x < e.

2) Mesh each layer's surface. Also draw x,y,z axes with logarithmically scaled tickmarks starting from the origin.

3) Cut out a big quarter slice of the whole sphere so that we can see its inside. Like this: https://scx1.b-cdn.net/csz/news/800/2019/magnetismdis.jpg

4) Apply a rotation to each layer, with the outermost layer completing a cycle in "1" unit time while the innermost layer completes a cycle in "e" (= 2.7) unit time.

5) Superpose on this first sphere a likewise onion layered second but Euclidean sphere. The latter's axes' tickmarks should be of a linear scale. Cut out too its quarter slice from some other angle to show its inside. Also apply uniform rotation to all its layers in similar fashion.

Answer 1

This should take care of your first and third requirements. I do not understand the others well, so I might wait for clarification.

SphericalPlot3D[
  Evaluate[E^Range[0, 1, 0.2]],
  {θ, 0, 3 Pi/2}, {ϕ, 0, 3 Pi/2}
]

Answer 2

Below is a recapitulation of the progress so far upon the suggestion of MarcoB. The code provided by MarcoB was modified as such:

Show[Graphics3D[
MapThread[{Black, Arrow@Tube@{{0, 0, 0}, #1}, 
Text[#2, #1, {0, -1}]} &, {3 IdentityMatrix[3], {x, z, y}}], 
Boxed -> False], 
SphericalPlot3D[
Evaluate[(E^2/2)*
Range[1, N[4 E], (N[4 E] - 1)/5]^(1/N[E])], {\[Theta], 0, 
Pi}, {\[Phi], 0, 3 Pi/2}], Axes -> True, 
AxesStyle -> {Red, Green, Blue}, FaceGrids -> All ]

Thus, it also includes x,y,z axes that both stick out from the origin AND envelop the 3D visual as a box. (They stick out alright, but alas, with no tickmarks as yet!)

Notice that I am still somewhat unsure about the "logarithmic scale ordering" of the concentric spheres. Because of the difficulty of seeing the axes starting from the origin, I had a hard time understanding the spaces between the concentric spheres. Eventually I think I managed to hand-input the correct thickness (Euler's number for the innermost sphere) that gets decreased to 1 as the radial distance goes to infinity.

Other items as part of the formulated question on top still remain. There is still the issue of rotating each of the concentric spheres and superposing on this setup another set of spheres that radiate out linearly.

Thanks!