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