(** 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_mult :
forall x y,
(x <> 0)%R -> (y <> 0)%R ->
(ln_beta x + ln_beta y - 1 <= ln_beta (x * y) <= ln_beta x + ln_beta y)%Z.