Ok I did the math, you need a 18mm f/3.1 on aps-c (27mm eq) to match the f1.8 lens in the OG RX100 (wide, 28mm eq, no zoom) if the same shutter speed is used on both cameras.
NO ... SAME (actual) f/stop (
on any sensor size) requires same Shutter-Speed and same ISO.
Assuming the goal is to make the same photo with different format cameras, the shutter speeds will be the same. However, the f-stops and ISOs will be different.
Two photos are equivalent if they're made from the same perspective, have the same angle of view & framing, the same depth of field, same rendering of movement, same noise visibility and same lightness. The sensors need to collect there same amount of light and the only way for different format sensors to capture the same total light is to work with different exposures.
So you are saying when i go buy a Sekonic/Gossen light meter .... i have to specify what format it is for ???
(learn something new every day)
Nobody is disputing that two different format cameras used in the same light and using the same f-stop, shutter speed and ISO to photograph the same scene will use the same exposure and make photos having the same lightness.
However there are significant ways in which those photos will or may be different. Due to the fact that the larger sensor has a larger surface area, it will collect more total light than a smaller sensor when both work with the same exposure. As a result, the image made with the smaller format camera will have greater noise visibility.
This alone means the photos made with the two cameras will not be the same. They won't be equivalent. The question of whether or not the increased noise visibility is objectionable, is subjective. It's up to the photographer to decide if that difference is acceptable or objectionable.
If different format cameras are used with the same exposure settings and ISO but different focal lengths to capture the same angles of view then, by definition, the smaller format camera will make a photo having greater depth of field. If two different focal length lenses are used at the same f-stop, the shorter focal length will have a smaller entrance pupil diameter. It will deliver an image having greater depth of field. This would also make the two photos not equivalent.
If we're using equivalence to demonstrate how it's possible for cameras of different formats to be used to make the same photo then, by definition, the cameras will work with different exposures to make those equivalent photos. It's not exposure that determines the amount of noise in a photo; it's the total light used that does so.
If we're using different format cameras to illustrate that the same exposure and ISO settings can be used to make photos having the same lightness regardless of format then, by definition, the photos will have different noise visibility. It's also possible they'll differ in other ways. Since different combinations of f-stop and shutter speed can deliver the same exposure to a sensor, it's possible that different format cameras working with the same exposure will make photos having the same lightness but different depths of field and rendering of movement in the frame.
Working with the same exposure not only guarantees that different format cameras will make different photos, it's possible the photos will be different in a multiple visually obvious ways.