Question

Define binary operations $\diamondsuit$ and $\heartsuit$ by

$a \diamondsuit b = a^{\log_7(b)} \quad \text{and} \quad a \heartsuit b = a^{\frac{1}{\log_7(b)}}$

for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ 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 $\log_7(a_{2019})?$

$8$

$9$

$10$

$11$

$12$

Solution:

Let $L(x) = \log_7 x.$ Then $L(a \diamondsuit b) = L(a)L(b)$ and $L(a \heartsuit b) = \dfrac{L(a)}{L(b)}.$

So $L(a_3) = \dfrac{L(3)}{L(2)},$ and $L(a_n) = \dfrac{L(n)}{L(n-1)} \cdot L(a_{n-1}).$ The product telescopes: $L(a_N) = \dfrac{L(3)}{L(2)} \cdot \dfrac{L(N)}{L(3)} = \dfrac{L(N)}{L(2)}.$

Hence $L(a_{2019}) = \dfrac{\log_7 2019}{\log_7 2} = \log_2 2019 \approx 10.98,$ which rounds to $11.$

Thus, the correct answer is D.

2019 AMC 12A Problem 23

Solution:

Problem 23 in Other Years