(** 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_sqrt :
forall x,
(0 < x)%R ->
(2 * ln_beta (sqrt x) - 1 <= ln_beta x <= 2 * ln_beta (sqrt x))%Z.