Is FACTORIZATION or PRIMES known to be in LOGSPACE Are the integer factorization and PRIMES known to be in LOGSPACE?

Recently, it has been shown by researchers that PRIMES is in P. But this does not say anything about LOGSPACE since it is not known if LOGSPACE = P.
Factoring is not known to be even in $\mathsf{P}$.

Primality is not known to be in any class conjectured to be smaller than $\mathsf{P}$ (AFAIK).