(** 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_div :
forall x y : R,
(0 < x)%R -> (0 < y)%R ->
(ln_beta x - ln_beta y <= ln_beta (x / y) <= ln_beta x - ln_beta y + 1)%Z.