Is plain multiplication a P problem? I am a CS novice who just recently encountered the P vs NP problem. And I was wondering whether plain multiplication is a P problem. If it is, does that mean there is a polynomial relationship between the increasing complexity and the time required to solve? I.e. Is there a polynomial relationship between the time required to solve x*1, x*2, x*3, x*4, etc?

I tried to get some "experimental results" by writing simple a Java program that multiplies BigIntegers and counts time of execution. Curiously, I noticed that the execution time only increased in a polynomial manner (e.g. 2ms -> 4ms -> 6ms, etc) when the "complexity" grew in a exponential manner (e.g. x*2 -> x*20 -> x*200, etc), meaning that time and execution of multiplication is a logarithmic relationship instead? Why is this?

Thanks in advance.
Yes, and with many different algorithms. 

See https://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations

One important thing to notice is that the size of the input is the number of digits in the numbers, rather than the magnitude of the numbers (e.g. the "size" of the number 657 is 3, rather than 657, because it has 3 digits).