Bowers' Arrays IV

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.