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2^n and 5^n have same starting digit

Source: SMO junior 2024 Q4

October 12, 2024
number theoryDigit

Problem Statement

Suppose for some positive integer nn, the numbers 2n2^n and 5n5^n have equal first digit. What are the possible values of this first digit?
Note: solved [url=https://artofproblemsolving.com/community/c6h312638p1685546]here
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