(** 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_mult_bpow :
forall x e, x <> 0%R ->
(ln_beta (x * bpow e) = ln_beta x + e :>Z)%Z.