If I add a Canon 1.4x III extender to my new EF 100-400mm f/4.5-5.6L lens, I believe I lose one stop of light. So what happens to the f/4.5-5.6 part? Does the f/5.6 part become f/8, for example, at the top end?
Yes. 5.6 becomes 8.0
If you add a 1.4x extender (teleconverter) you will now have a 140-560mm f/6.3-f/8 lens.
Keep in mind that most cameras will only autofocus to a limit of f/5.6 and f/8 will be too dark for the AF to work.
Newer high end cameras like the 7D Mk II and 5D Mk III will attempt to AF with f/8 lenses. Even if AF works (as in "the camera will try to AF") at f/8, it will be slower, less accurate, will need brighter light and/or a higher contrast target upon which to focus, and generally be restricted to fewer usable AF points. The 7DII and 5DIII will only attempt to AF at f/8 with the center focus point selected. The new 1D X Mark II is the only Canon camera that can use every AF focus point at f/8.
For more on the practical ramifications of using teleconverters with slower lenses, please see Will the Canon 5D MK II with 100-400 1:4.5-5.6 work properly with Kenko 1.4 converter?
The f/number is used to gauge the light passing power of a lens. The camera lens projects an image of the outside world onto the surface of the digital imaging chip (or film). The key factors are: Scene brightness, the sensitivity (ISO), the working dimeter of the lens (aperture), and the focal length.
The f/number calculation is the focal length divided by the working diameter of the aperture. As an example, if the focal length is 200mm and the working diameter is 25mm, the f/number is 200 ÷25=8 written as f/8. The f/ is shorthand for “focal ratio”.
The f/number involves the interweaving of focal length and aperture diameter. Why? The camera lens acts like a funnel in that it gathers light. The larger the surface area of the opening, the lighter gathered, the brighter the projected image. The focal length decrees the size of the image projected. As focal length increases, the image size is magnified. A magnified image is spread over more area. Consider a movie of slide project image. As we increase the distance lens to screen, the image gets bigger thus it is forced to cover more surface. The enlarged image is thus dimmer. The loss of light is 4X when the focal length is doubled. This is because the size of objects will enlarge 2X however the surface area the image increases 4X.
A 400mm lens is operating at f/5.6; the working diameter is 71.4mm. Mount a 1.4X extender and the focal length lengthens to 400 X 1.4 = 560mm. The working dimeter did not change, the f/number = 560 ÷ 71.4 = 7.8 written as f/7.8. This is slightly more loss than 1 stop.