I need help determining the sample size necessary for a logistic regression (binary DV) with two continuous predictors and also the sample size necessary if I use three continuous predictors.


tl;dr For a binary response model (e.g. logistic regression) you need about 15*(# of non-intercept parameters) successes or failures (whichever is less). Estimating a simple linear effect of a continuous predictor takes a single parameter. So in your case you need $\textrm{min}(\textrm{successes},\textrm{failures}) > 30$ (for 2 predictors) or $>45$ (for 3 predictors).

This is laid out clearly in F. Harrell's Regression Modeling Strategies (Springer), pp. 72-73 (you should buy or borrow the book and read the whole thing, but especially Chapter 4):

in many situations a fitted regression model is likely to be reliable when the number of predictors ... $p$ is less than $m/10$ or $m/20$, where $m$ is the "limiting sample size" given in Table 4.1

And here's Table 4.1 (Google Books screenshot)

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