(** 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
}.(** 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.