(** 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.

Lemma ln_beta_plus :
forall x y,
(0 < y)%R -> (y <= x)%R ->
(ln_beta x <= ln_beta (x + y) <= ln_beta x + 1)%Z.