3.31: Logarithmic AP inequality and 25th term

The first three terms of an arithmetic progression can be written as $2\mathrm{ln}(x^{3})$, $5\mathrm{ln}(x^{2})$, $2\mathrm{ln}(x^{7})$, where $x>1$. Find the least number of terms for the sum of this progression to be greater than $1032\mathrm{ln}\,x$.

Given the $25$th term equals $408$, find the exact value of $x$.