Consider the following question, does there exist a power of $2$ which begins with the digits $7777$? If so, are there infinitely many powers of $2$ that begin with the digits $7777$? The answer to this question, surprisingly, has a solution using dynamical systems; that is, we consider repeated application of a function $T$ from a space $X$ to itself, $T:X\rightarrow X$. This talk has three goals. First, we will introduce the field of dynamical systems. Second, we will use dynamical systems to answer the posed question. Third and finally, we will relate this question to Benford's Law and discuss some of its applications.