Need some help regarding calculations involving natural logarithm

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I wanted to calculate the values of $\log_e{10^n}$ for $n=1,2 \dots 30$.

I tried simplifying $\log_e{10^n}$ as $n\log_e{10}$. It can be re-written as $n\cdot 2.3025$ but I am getting inaccurate results.

I tried using calculator but it is showing different values for $\log_e{10^n}$ and $n\log_e{10}$ which is not accurate. I don't have any knowledge on Mathematica. Can someone help me with finding these values?
I want that $|\log_e n-a_n| \leq 10^{-3}$ where $a_n$ are the computed values for $n=1 \dots 30$. You can suggest ways