How to solve mixed power and exponential equation?

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The exercise says that I should find "real zeros" of the following function:

$$f(x)=7-2x^2+2^{-x}$$

I have tried several ways, but I keep getting stuck as I take log-s of both sides. Can I solve this equation manually, somehow, not in a very complicated way?

(I suspect, it shouldn't be very hard, since it's in the introductory chapter of the calculus book :/)