Question 23
Define binary operations $\diamondsuit$ and $\heartsuit$ by
$$
a \diamondsuit b = a^{\log_7(b)}
$$ and $$
a \heartsuit b = a^{\frac{1}{\log_7(b)}}
$$
for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3 = 3 \heartsuit 2$ and $$
a_n = (n \heartsuit (n - 1)) \diamondsuit a_{n-1}
$$ for all integers $n \ge 4.$ To the nearest integer, what is $\log_7(a_{2019})?$
$$
a \diamondsuit b = a^{\log_7(b)}
$$ and $$
a \heartsuit b = a^{\frac{1}{\log_7(b)}}
$$
for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3 = 3 \heartsuit 2$ and $$
a_n = (n \heartsuit (n - 1)) \diamondsuit a_{n-1}
$$ for all integers $n \ge 4.$ To the nearest integer, what is $\log_7(a_{2019})?$