Define binary operations $♢$ and $♡$ by

$a♢b=a^{{\log }_{10}(b)}\text{and}a♡b=a^{\frac{1}{{\log }_{10}(b)}}$

for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3=3♡2$ and

$a_n=(n♡(n-1))♢a_{n-1}$

for all integers $n\ge 4$. To the nearest integer, what is ${\log }_{10}(a_{2024})$?