Logarithm Problem: Finding Sum of Non-Negative Integers

This problem asks us to find the value of \(a + b + c\) given that \(a\), \(b\), and \(c\) are non-negative integers, and their logarithms sum to zero. The given equation is \(\log a + \log b + \log c = 0\).

Logarithm Properties Application

To solve this, we first need to use a fundamental property of logarithms. The sum of logarithms with the same base is equal to the logarithm of the product of their arguments. This property is expressed as:

• \(\log_B M + \log_B N = \log_B (MN)\)

Applying this property to our given equation \(\log a + \log b + \log c = 0\), we can combine the terms on the left side:

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

Determining the Product of Integers (abc)

Now, we need to consider what it means for a logarithm to be equal to zero. For any valid logarithm base \(B\) (where \(B > 0\) and \(B \neq 1\)), if \(\log_B X = 0\), then the argument \(X\) must be equal to 1. This is because any non-zero number raised to the power of 0 is 1 (i.e., \(B^0 = 1\)).

Therefore, from the equation \(\log (a \cdot b \cdot c) = 0\), we can conclude that the product of \(a\), \(b\), and \(c\) must be 1:

\(a \cdot b \cdot c = 1\)

Finding the Values of a, b, c

We are given that \(a\), \(b\), and \(c\) are non-negative integers. This means they can be \(0, 1, 2, 3, \ldots\). We need to find values for \(a\), \(b\), and \(c\) that satisfy both conditions: they are non-negative integers, and their product is 1.

Let's analyze the possibilities:

• If any of \(a\), \(b\), or \(c\) were \(0\), their product \(a \cdot b \cdot c\) would be \(0\), not \(1\). Thus, \(a, b, c\) cannot be \(0\).
• Since they must be integers and their product is \(1\), the only positive integer solution is if each integer itself is \(1\).

Therefore, the only possible integer values for \(a\), \(b\), and \(c\) that satisfy \(a \cdot b \cdot c = 1\) and are non-negative are:

• \(a = 1\)
• \(b = 1\)
• \(c = 1\)

These values also ensure that the original logarithm expression \(\log a + \log b + \log c\) is defined, as logarithms are defined only for positive arguments (\(X > 0\)). Since \(1 > 0\), \(\log 1\) is defined (and is equal to 0).

Calculating the Sum \(a + b + c\)

Now that we have determined the individual values of \(a\), \(b\), and \(c\), we can find their sum:

\(a + b + c = 1 + 1 + 1\)

\(a + b + c = 3\)

Thus, the value of \(a + b + c\) is 3.