Can the discret log problem be solved when the modulus is a hard to factor composite number, i.e. when modulus $n=p*q$, where $p$ and $q$ are two large prime numbers?
The problem is solving for $x$ the equation $g^x\bmod n=a$, given integers $n$, $g$, $a$, with $n$ a large composite too difficult to factor (which implies $n$ is several hundreds bits). It is not stated how instances of the problem are generated.
That is hard at least for some instances of the problem. In particular, if one of the factor $p$ of $n$ is large (say, at least 1536-bit), and $p-1$ has a large factor $p_0$ (say, at least 256-bit), and the order of $g$ in $\Bbb Z_p^*$ is $p_0$ or a larger multiple, then an instance of the problem where $x$ was chosen at random in $[1,n/2]$ and $a$ computed as $g^x\bmod n=a$ is hard. Argument: ability to solve $g^x\bmod n=a$ for $x$ and knowledge of the factorization of $n$ trivially implies ability of solving $g^x\bmod p=a\bmod p$ for $x$ (modulo $p-1$), which is is believed hard; and lacking knowledge of the factorization of $n$ can only make the question's problem harder.
I conjecture without proof that the condition "$p$ large" can be replaced by the existing "$n$ too difficult to factor"; and that a random instance is hard.
Among the usual Discrete Logarithm methods, Baby-step/Giant-step and Pollard's Rho remain applicable, with cost $O(\sqrt x)$ multiplications modulo $n$ (and moderate memory for Pollard's Rho). That allows to solve instances with moderate $x$ (say 80-bit with a standard PC), but I see no better method to solve a random instance.