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

The correct option is -3 and 3.

Let's solve the problem step-by-step to understand how to arrive at this answer.

We are given the logarithmic equation:
$\log \left|a\right|+\log \left|b\right|+\log \left|c\right|=0$
where $a$, $b$, and $c$ are integers.

First, recall the product rule of logarithms, which states that:
$\log \left(x\right)+\log \left(y\right)+\log \left(z\right)=\log (x·y·z)$
Applying this property to our equation, we get:
$\log \left|a·b·c\right|=0$

By converting this logarithmic equation to its exponential form (regardless of the base of the logarithm, since any non-zero base raised to the power of 0 equals 1), we obtain:
$\left|a·b·c\right|=1$

This can also be written as:
$\left|a\right|·\left|b\right|·\left|c\right|=1$

Since $a$, $b$, and $c$ are integers, their absolute values $\left|a\right|$, $\left|b\right|$, and $\left|c\right|$ must also be integers. The only way the product of three positive integers equals 1 is if each individual integer is equal to 1.
Therefore:
$\left|a\right|=1,\left|b\right|=1,\text{and}\left|c\right|=1$

This gives us the possible values for each variable:
$a=\pm 1$
$b=\pm 1$
$c=\pm 1$

Now, we find the minimum and maximum values of the sum $a+b+c$:

1. Finding the Maximum Value:
To maximize the sum, we choose the largest possible values for $a$, $b$, and $c$, which are $a=1$, $b=1$, and $c=1$.
${\text{Maximum Sum}}_{}=1+1+1=3$

2. Finding the Minimum Value:
To minimize the sum, we choose the smallest possible values for $a$, $b$, and $c$, which are $a=-1$, $b=-1$, and $c=-1$.
${\text{Minimum Sum}}_{}=-1+(-1)+(-1)=-3$

Thus, the minimum and maximum values of $a+b+c$ are -3 and 3 respectively.