Most important feature of a telescope I was taking an astronomy test, the following question came up:


  The most important function of the astronomical telescope is:
  
  
  Resolving power
  Light gathering power
  magnifying power
  focal length
  


The focal length obviously means nothing here, as well as the magnification (that can always be done later). As far as resolving power and light gathering power go though, they both seem to be "equally" important, in that without the first, there is no detail, and without the second, there is no image.

It does seem that exposure time can compensate for a lack of light gathering power (what matters is the product between the two), but nothing can compensate for resolving power.

My professor seems to think the answer is resolving power. Why is this? Is this some well known idea in astronomy?
Well the angular resolution of a telescope depends on two things: 1) The wavelength of light you're looking in, and 2) the diameter of the primary lens/mirror. 

$$ \theta = 1.22 \frac{\lambda}{D}$$

where $\lambda$ is the wavelength of light and $D$ is the diameter of the lens.

Light collecting power is simply how many photons you can collect in a particular wavelength in a given amount of time. Essentially, the bigger the bucket (or lens/mirror), the more rain you can catch. You can sort of get around light collecting power by increasing your exposure time. But, if the size of the object you're trying to observe is smaller than your telescope's resolving power, I don't believe there is really much you can do about it. Being limited by the resolution of your telescope is called being diffraction limited, but most telescopes on Earth are usually seeing limited due to the atmosphere's smearing effects.