AMC 12A 2019 Problem 23

Define binary operations \diamond and \heartsuit by

a \diamond b=a^{\log _{7}(b)} \quad \text { and } \quad a \heartsuit b=a^{\frac{1}{\log _{7}(b)}}

for all real numbers a and b for which these expressions are defined. The sequence \left(a_{n}\right) is defined recursively by a_{3}=3 \heartsuit 2 and

a_{n}=(n \heartsuit(n-1)) \diamond a_{n-1}

for all integers n \geq 4. To the nearest integer, what is \log _{7}\left(a_{2019}\right)?

Answer Choices
A. 8
B. 9
C. 10
D. 11
E. 12