(** Another well-used function for having the logarithm of a real number x to the base #&beta;# *)
Record ln_beta_prop x := {
ln_beta_val :> Z ;
_ : (x <> 0)%R -> (bpow (ln_beta_val - 1)%Z <= Rabs x < bpow ln_beta_val)%R
}.
Definition ln_beta :
forall x : R, ln_beta_prop x.

Theorem ln_beta_unique_pos :
forall (x : R) (e : Z),
(bpow (e - 1) <= x < bpow e)%R ->
ln_beta x = e :> Z.