Does there exists a positive integer $n $ such that the decimal representation of $3^n$ ...

This question was asked in a masters entrance examination and I am not able to make any significant progress on this problem.

Question: Does there exists a positive integer $n$ such that the decimal representation of $3^n$ starts with the digit 2019? Justify your assertion.

I have been following the textbook Introduction to Analytic number theory by Tom M Apostol.

The factorization of $2019$ is $3\times 673$ where it can be seen that $673$ is prime as it is not divisible by any number less than $\sqrt{673}$. But I am not able to move forward from this using the results I know based on $1$st course on Number Theory.

Can you please help me with the problem.

I shall be very grateful.