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 $a _ { n } = 10 ^ { n \times \text{(나)} }$.

Let $f ( n )$ and $g ( n )$ be the expressions that fit in (가) and (나), respectively. What is the value of $\frac { g ( 10 ) } { f ( 4 ) }$? [4 points]\\
(1) 38\\
(2) 40\\
(3) 42\\
(4) 44\\
(5) 46