Is there already a clipping mask applied in the file?
Then drawing inside doesn't work that easily.
You can double click your logo to get into isolation mode. Then replace whatever is already inside.
As I (mis)understand it, what you need is to show inside rather than draw inside.
You can just place the image beneath the text, which can be live Type, then Ctrl/Cmd+7 to create a Clipping Mask, which will hide everything in the image outside the latters of the text.
With the Clipping Mask Group expanded in the Layers palette, you can (deselect and) Direct Select the image and move it about for best looking fit of the visible parts under the letters.
If you already have (taken over an AI file with) a Clipping Mask containing an older image, you can just place the new image and drag it down into the expanded Clipping Mask Group right above the older image and delete or hide the latter, or just leave it be (by hiding/leaving it, you can use it again by hiding the new one (and showing the old one if hidden), and that way you can switch back and forth between images).
You can also extend the switching option by gathering whichever number of images desired in the Clipping Mask and switching between their being visible.
As a silly afterthought, I believe a Clipping Mask could actually also have been called Show Inside; I prefer Clipping Mask though, for a few reasons.