One of the important activities in photography, light metering tells you about the amount of light that falls on the scene - or the amount of light reflected towards the camera.
The result of light metering is used to decide a proper exposure - since there are many couples of f stop/exposure time that will give the same darkening of the film.

In some cases you don't even need a light meter, either because the lighting situation is well known or because you are such an expert that you can guess exposure. When using flashes of known power as the main light source, you can dispense with light metering and use a simple measuring tape.

Light metering is done with a light meter (duh !). Some light meters can measure strobe light, and they are called flashmeters.

Light metering becomes mildly complex when the photographer mixes continuous light (like daylight, incandescent lightbulbs) with electronic flashes.
In this case you have to keep in mind that

continuous light is controlled via exposure time and f stop
flash light is controlled via f stop alone

the first fact is quite clear, but what about the second ? The reason is that an electronic flash is so fast - generally faster than 1/1000 of a second, that it does not matter if you keep the shutter open for one second of for 1/100th of a second: the film is going to get all the strobe's light anyway.
Let us have an example of light metering in mixed light. Assume a fairly dumb manual camera, or a view camera, and a fixed-output studio flash.

You have a woman sitting in front of a window. Far behind her, unlit by the window, you have a white wall. The woman is lit by the window, and the white wall you will light with your flash. You want that background wall to be white.
You meter a grey card, and you realize that the woman needs to be exposed at 1/30 8.
You place your studio flash 2 meters away from the wall, and you realize that, due to its guide number the correct exposure for something 2 meters away is f 4.
What can you do ?
  1. You can say "aw shucks". The picture will have a grayish background. You lose.
  2. You can use your head. Dial f 4 on your camera (2 stops, that's to say four times as much light coming in), and change the exposure time to four times less: 1/30 / 4 = 1/120, in fact 1/125. This way you the woman is properly exposed, and the background becomes white.
This technique is fundamental for fill flash. For more mixed light insanity, see color shift.

Other relevant nodes: automatic exposure, matrix metering, center weighted metering, spot meter, light meter, TTL.

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