To determine the minimum and maximum values of a + b + c for integers a, b, c, given the logarithmic equation log |a| + log |b| + log |c| = 0, we will first simplify the given expression using logarithm properties.

Logarithm Equation Simplification

The sum of logarithms can be expressed as the logarithm of the product of their arguments. This is a fundamental property of logarithms:

$\log X + \log Y + \log Z = \log (XYZ)$

Applying this property to our given equation, log |a| + log |b| + log |c| = 0, we get:

$\log (|a| \cdot |b| \cdot |c|) = 0$

Absolute Product Derivation

Next, we utilize the property that if $\log M = 0$, then the argument $M$ must be equal to 1. This holds true for any valid logarithm base, as any non-zero number raised to the power of 0 equals 1.

From $\log (|a| \cdot |b| \cdot |c|) = 0$, we deduce:

$|a| \cdot |b| \cdot |c| = 1$

Integer Value Determination for a, b, c

We are given that a, b, c are integers. For the product of their absolute values ($|a|$, $|b|$, $|c|$) to be equal to 1, and since absolute values are non-negative, each individual absolute value must be 1. This is because 1 is the only positive integer whose only positive integer factor is itself.

Therefore, we must have:

• $|a| = 1$
• $|b| = 1$
• $|c| = 1$

This implies that each integer a, b, c can only take one of two possible values:

• $a = 1$ or $a = -1$
• $b = 1$ or $b = -1$
• $c = 1$ or $c = -1$

Maximum Value of a + b + c

To obtain the maximum possible value for the sum a + b + c, we should choose the largest possible value for each of the variables a, b, c. From our derivation, the largest possible integer value for each variable is 1.

• Choose $a = 1$
• Choose $b = 1$
• Choose $c = 1$

The resulting maximum sum is:

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

Minimum Value of a + b + c

Conversely, to find the minimum possible value for the sum a + b + c, we should select the smallest possible value for each of the variables a, b, c. From our derivation, the smallest possible integer value for each variable is -1.

• Choose $a = -1$
• Choose $b = -1$
• Choose $c = -1$

The resulting minimum sum is:

$a + b + c = (-1) + (-1) + (-1) = -3$

Summary of Results

Based on these calculations, the minimum and maximum values of a + b + c are -3 and 3 respectively.

Therefore, the minimum value is -3 and the maximum value is 3.