2016 AMC 10A Problem 2

Rewrite all bases as powers of $10$: $$\small 10^x\cdot100^{2x}=10^x\cdot(10^2)^{2x}=10^{5x},$$ and $$1000^5=(10^3)^5=10^{15}.$$ Therefore $5x=15$, so $x=3$. Thus, the correct answer is C.