<s>[INST] If $7$ is the first digit of $2^n$, what is the first digit of $5^n$? Let $2^n = 7\cdot 10^x + p$ and $5^n = a\cdot 10^y + r$
And now what?
(We're in base $10$)
 [/INST] By assumption, we can write $2^n=x\cdot 10^m$ with $7\le x<8$.
Also, we can write $5^n=y\cdot 10^k$ with $1\le y<10$.
Then $10^n=2^n5^n=xy10^{m+k}$, so $xy$ must be a power of $10$.
If $y\ge 2$, we obtain $14\le xy<80$, no power of ten. Therefore $1\le y<2$, i.e. the leading digit of $5^n$ is $1$.
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