Bowers' Arrays IV

Number Theory Level pending

For this problem, \({ log }_{ 3 }^{ x }(n)\) denotes the repeated \(log_{ 3 }\). For example, \({ log }_{ 3 }^{ 2 }(n)\) represents \({ log }_{ 3 }({ log }_{ 3 }(n))\).

With that in mind, what is the sum of \(a\) and \(b\) in the equation for \(x\) so that \({ log }_{ 3 }^{ x }(\{ 3,n,2\} )\) returns a decimal number with fewer than ten digits, when the equation is written in the form \(x = an + b\)?

Note: Bowers' Arrays are a fairly obscure notation. An explanation of them can be found here.

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