A sequence $\left\{ a _ { n } \right\}$ with all positive terms has $a _ { 1 } = 10$ and satisfies $$\left( a _ { n + 1 } \right) ^ { n } = 10 \left( a _ { n } \right) ^ { n + 1 } \quad ( n \geq 1 )$$ The following is the process of finding the general term $a _ { n }$.
Taking the common logarithm of both sides of the given equation, $$n \log a _ { n + 1 } = ( n + 1 ) \log a _ { n } + 1$$ Dividing both sides by $n ( n + 1 )$, $$\frac { \log a _ { n + 1 } } { n + 1 } = \frac { \log a _ { n } } { n } + ( \text { (가) } )$$ Let $b _ { n } = \frac { \log a _ { n } } { n }$. Then $b _ { 1 } = 1$ and $$b _ { n + 1 } = b _ { n } + \text { (가) }$$ Finding the general term of the sequence $\left\{ b _ { n } \right\}$, $$b _ { n } = \text { (나) }$$ Therefore, $$\log a _ { n } = n \times \text { (나) }$$ Thus $a _ { n } = 10 ^ { n \times ( \text { (나) } ) }$.
When the expressions for (가) and (나) are $f ( n )$ and $g ( n )$ respectively, what is the value of $\frac { g ( 10 ) } { f ( 4 ) }$? [3 points]
(1) 38
(2) 40
(3) 42
(4) 44
(5) 46