Attacks against El Gamal private key El Gamal encryption involves picking $(p,g,b)$ which is our public key. We compute $b=a^x$ $mod$ $p$. Here, $x$ is the private key which we don't know. 

What are some efficient and strong algorithms today used to finding this $x$? 

I am currently dealing with numbers such as $b=42-60$ digits long and $p=30-50$. So $b$ is anywhere between 42 and 60 digits. 

Does anyone know of any program and some attacks to finding this $x$ using our given information?
Being new to cryptography is one thing, but you are supposed to do some research on your own before asking questions here (see How to Ask), and D.W. gave you the right directions already.

But since you wanted names and links:


The first stop should be discrete logarithm on Wikipedia, and it lists several algorithms on this topic.
As a beginner, start with Babystep-Giantstep and Pohlig-Hellman.
Your next stop should be Index Calculus.
Additionally, there are algorithms based on the NFS, e.g. described in Gordon's paper Discrete Logarithms in GF(p) Using the Number Field Sieve. Current records modulo primes employ this technique.