I was just reading an anecdote about a third-grade student who was asked by her math teacher to find a number which, when two is raised to the power of that number, produces a number that starts with the digit 7 (in base 10, as the context makes clear).
I was able to quickly write a one-liner in Perl that found the smallest such number, 46, which is also the number ultimately found by the student. But the story states that the question was given in a math class, and implies that the student found the answer without the aid of a computer.
I haven't been able to come up with an avenue of attack for this problem that doesn't depend on a computer, much less one that a third-grader, even at a relatively advanced level of mathematics, might discover. So how can this be done?
In the more general case, it would be nice to see a solution to the problem of "Given an integer $b$ and digit $d$, what's (a/the smallest) power $n$ such that $b^n$ has leading digit $d$ in base 10?"