Consider the equation $x^6 = 6^x$, where $x$ is a real number. It is easy to see that $x = 6$ is a solution, since both sides are then $6^6$. A class is asked whether there are any others. One student says that a power of $x$ grows more slowly than an exponential, so the two sides can be equal only once. Another says that a negative $x$ cannot work because $6^x$ is then less than 1. Count every real solution, whether positive, zero or negative. You may use $2 < e < 3$. How many real numbers $x$ satisfy $x^6 = 6^x$?