MathDB

Define binary operations

\diamondsuit

and

\heartsuit

a \, \diamondsuit \, b = a^{\log_{7}(b)} \qquad \text{and} \qquad 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 \geq 4

. To the nearest integer, what is

\log_{7}(a_{2019})

<span class='latex-bold'>(A) </span> 8 \qquad <span class='latex-bold'>(B) </span> 9 \qquad <span class='latex-bold'>(C) </span> 10 \qquad <span class='latex-bold'>(D) </span> 11 \qquad <span class='latex-bold'>(E) </span> 12

Problem Statement