(** 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.

Theorem bpow_ln_beta_le : forall x, (x <> 0)%R -> (bpow (ln_beta x-1) <= Rabs x)%R.