Solving for Minimum and Maximum Integer Sum

The problem asks for the minimum and maximum values of the sum $a + b + c$, where $a$, $b$, and $c$ are integers satisfying the equation $\log |a| + \log |b| + \log |c| = 0$.

Understanding the Logarithm Condition

Using the properties of logarithms, we can simplify the given equation:

• The sum of logarithms is the logarithm of the product: $\log (|a| \cdot |b| \cdot |c|) = 0$
• Exponentiating both sides with base 10 (or simply using the definition of logarithm): $|a| \cdot |b| \cdot |c| = 10^0$
• Therefore, the condition simplifies to: $|a| \cdot |b| \cdot |c| = 1$

Determining Integer Constraints

Since $a$, $b$, and $c$ are integers, their absolute values $|a|$, $|b|$, and $|c|$ must be positive integers. The only way the product of three positive integers can equal 1 is if each integer is equal to 1.

• $|a| = 1 \implies a = 1$ or $a = -1$
• $|b| = 1 \implies b = 1$ or $b = -1$
• $|c| = 1 \implies c = 1$ or $c = -1$

Calculating Minimum and Maximum Sums

To find the minimum value of $a + b + c$, we choose the smallest possible integer values for $a$, $b$, and $c$.

• Minimum sum: Set $a = -1$, $b = -1$, $c = -1$. $a + b + c = (-1) + (-1) + (-1) = -3$

To find the maximum value of $a + b + c$, we choose the largest possible integer values for $a$, $b$, and $c$.

• Maximum sum: Set $a = 1$, $b = 1$, $c = 1$. $a + b + c = 1 + 1 + 1 = 3$

The minimum value is -3 and the maximum value is 3.