Can I solve for $\sum_{i=1}^{n}{a_i}$ knowing $\sum_{i=1}^{n}{\ln(a_i)}$?

Question

My friend posed this problem to me that I have no idea how to solve.
I know $\sum_{i=1}^{n}{\ln(a_i)}$, so let $$\sum_{i=1}^{n}{\ln(a_i)}=x$$ So, in terms of $x$, what is $\sum_{i=1}^{n}{a_i}$?

Answer 1

No, because $\prod_ia_i$ does not determine $\sum_i a_i$ (note $\sum_i\ln a_i = \ln\prod_i a_i$, so you are essentially given $\prod_ia_i$).

For example, $x\cdot\frac1x = 1$ for any positive $x$, but $x + \frac1x$ can be many different things depending on what $x$ you pick.

(In the language of the question, you would be given $\ln x + \ln\frac1x = 0$, which is equivalent to $x\cdot \frac1x = 1$).

Answer 2

No.

$\ln 1 + \ln 6 = \ln 2 + \ln 3$ but $1+6 \not= 2+3$.