(** 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 ln_beta_le_Zpower :
forall m e,
m <> Z0 ->
(Zabs m < Zpower r e)%Z->
(ln_beta (Z2R m) <= e)%Z.