How to find the number of positive integral points (points that are positive integers) that satisfy a given exponential function?

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For example if I have an equation f(x) =1/(2^x -3), by intuition found for x=2 , f(x)=1 . Then If we plot the graph of f(x) we find that 2,1 lies on the graph of the function. When x>2, f(x) decreases and has an asymptotic y=0 . When x<2 the graph has vertical asymptotic x=ln(3)/ln(2). Now I'd we have an exponential function of this type with n variables like f(x1,x2...,xn-1) how can we find number of integral points lying on the graph.