Compute discrete logarithm mod $n=p \times q$ knowing factorisation I read in a document that for a given $n = p\times q$ ($p$, $q$ primes), if you know $p$ and $q$ then you can easily solve the discrete logarithm problem, i.e. for fixed $a,b$, you can find $x$ such that 
$$a^x = b \bmod n$$
But I don't see how this can be done... I tried to check if the Chinese reminder theorem could help me, and this reduces to find the discrete logarithm modulo $p$ and modulo $q$, but I still don't see how we can get this new discrete logarithm without bruteforcing (which is still exponential, even if you get a square root improvement over the naive bruteforce).

Thank you!
This is not true in general. Bach showed that given the factorization of $n$, computing the discrete log modulo $n$ requires computing it modulo the factors (also cf. this previous question for more details.). So, if both the factors are large then DLP is still hard.