What are the minimum and maximum values of $a+b+c$, given $\log|a|+\log|b|+\log|c|=0$?

Question

For integers $a$, $b$ and $c$, what would be the minimum and maximum values of $a+b+c$ if $\log|a|+\log|b|+\log|c|=0$?

Given possibilities:
$(a)\ -3\ \text{and}\ 3\quad $ $(b)\ -1\ \text{and}\ 1\quad $ $(c)\ -1\ \text{and}\ 3\quad $ $(d)\ 1\ \text{and}\ 3\quad $

My trial:

$$\log|abc|=\log1\\ |abc|=1\\ abc=1 \text{ or } abc=-1$$ For maxima of $a+b+c$ we know $a=b=c$ then $a^3=1\implies a=1=b=c$

$\therefore a+b+c=1+1+1=3$

For minima of $a+b+c$ we know $a=b=c$ then $a^3=-1\implies a=-1=b=c$

$\therefore a+b+c=-1-1-1=-3$.

Thus the correct option is (a). Please explain if I am wrong. Is there another, easier way to solve this?

Answer 1

You are correct. I'm fairly certain that there in no easier way to solve this.