jmc

If you notice that 340 is close to $7^3=343=1000_7$, then it makes the problem much easier. Since $343=1000_7$, that makes $342=666_7$, or the largest three-digit number in base 7. To get 340, we subtract 2, which means subtracting 2 from the digit in the $7^0$ place. That leaves us with $\boxed{664_7}$.

Alternatively, the largest power of 7 less than 340 is $7^2=49$, and the largest multiple of 49 less than 340 is $6\cdot 49=294$. That means there is a 6 in the $7^2$ place. We have $340-294=46$ left, which can be represented as $6\cdot 7^1+4\cdot 7^0$. So, we get $340=6\cdot 7^2+6\cdot 7^1+4\cdot 7^0=\boxed{664_7}$.