(** 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_lt_pos : forall x y, (0 < y)%R -> (ln_beta x < ln_beta y)%Z -> (x < y)%R.