2019 AMC 12A Problem 23

Question

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})?

8

9

10

11

12

Apply

\log_7:

then

\log_7(a \diamondsuit b) = \log_7 a \cdot \log_7 b

Writing

L(x) = \log_7 x,

the recurrence becomes

L(a_n) = L(a_{n-1}) \cdot \dfrac{L(n)}{L(n-1)}

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:

\begin{aligned} L(a_N) &= \dfrac{L(3)}{L(2)} \cdot \dfrac{L(N)}{L(3)} \\ &= \dfrac{L(N)}{L(2)}. \end{aligned}

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.

Problem 23 in other years