We extend Waring’s problem to a wide class of functions. This wide class of functions include Hardy’s logarithmic exponential functions of polynomial growth, ie functions we can get by using the symbols log, exp, the real variable x, real constants, addition and multiplication. One of the consequences of our result is that if a(x) is a logarithmic exponential function of polynomial growth, then the integer parts [a(1)], [a(2)], [a(3)] . . . always form an asymptotic basis for the natural numbers unless a(x) is a rational polynomial with gcd([a(1)], [a(2)], [a(3)] . . .) > 1.