jmc

We first find that the largest power of $2$ that is less than $84$ is $2^6=64$. Our next step is to find the largest power of $2$ that is less than $84-64=20$ which is $2^4=16$. This leaves us left with $20-16=4$, but $4=2^2$, so we have $$84=1\cdot 2^6+0\cdot 2^5+1\cdot 2^4+0\cdot 2^3+1\cdot 2^2+0\cdot 2^1+0\cdot 2^0.$$Thus, our base $2$ representation of $84_{10}$ is $\boxed{1010100_2}$.