First Three Digits of Powers of 2 and 5 Suppose you know that there exists positive integer $n\in \mathbb{N}$ such that the first three digits of $2^n$ and $5^n$ are the same, and that $\forall n$ that do so, the first three digits are unique. Then what are those first three digits?
Edit: First three digits from left to right.
 A: If two numbers share the same first three digits, then their ratio must be very nearly a power of ten (within 0.1% or so).  That is, $(5/2)^{n} \approx 10^k$, or
$n\log(5/2)\approx k\log 10$, or
$$
\frac{k}{n}\approx \log_{10}(5/2) \approx 0.39794001
$$
for natural numbers $k$ and $n$.  So we search for rational approximants to this number, and find
$$
\frac{425}{1068} \approx 0.39794007,
$$
which agrees to six places.  And indeed the first four digits of both $2^{1068}$ and $5^{1068}$ are the same: $3162$.

To find this without searching, follow up on @GerryMyerson's hint.  If $2^n\approx \alpha 10^{a}$, and $5^n\approx \alpha 10^{b}$, for some number $\alpha$, then $\alpha^2 \approx 10^{k}$, and so (if $k$ is odd) the first three digits of $\alpha$ are the same as the first three digits of $\sqrt{10}=3.16227766\ldots$.  (If $k$ is even, then one of $2^n$ and $5^n$ will be just above a power of $10$ and the other will be just below a power of $10$, so their initial digits will differ.)
