To prove that there is no value of k between 12 and 10^10 such that 2^k has all even digits, you only have to prove that there is an odd digit among the lowest X decimal digits for all 12 ≤ k ≤ 10^10.

The value of X necessary to prove this grows rather slowly compared to k. For example, the smallest power of 2 that doesn't have an odd digit in its last 16 digits is 2^12106. The smallest power of 2 that doesn't have an odd digit in its last 32 digits is 2^3789535319. So it makes sense to try increasingly large values of X until you are able to rule out all values of 2^k for k up to 10^10.

Here's a C++ program you can run to replicate this proof. It takes around 20 minutes to run, and can probably be optimized further, but it shows the principle: https://pastebin.com/DVK2JKdq