91Ó°ÊÓ

Understanding the domain of a logarithmic function involves knowing the restrictions that apply to its arguments. In essence, a logarithmic function, expressed as \( f(x) = \log(a(x)) \), requires the input \( a(x) \) to be strictly positive. This is because the logarithm of a positive number is defined, while the logarithm of zero or a negative number is undefined in real number mathematics.

When dealing with a function like \( f(x) = \log \left(\frac{1}{x-[x]}\right) \), we must ensure that the denominator, \( x-[x] \), is positive to satisfy this key constraint. This demands us to think about the behavior of \( x \) in relation to its integer part \( [x] \). By imposing the condition that the argument of the logarithm must be greater than zero, you establish the foundation of determining the domain for such functions.

Dealing with inequalities is a significant part of identifying the domain of functions, where certain inputs result in undefined or nonreal outputs. In our exercise, we have the inequality \( \frac{1}{x-[x]} > 0 \), originating from the restriction of the logarithmic function. To solve this, you analyze the behavior of the denominator: \(x-[x]\).

This fraction expresses the non-integer part of \(x\), which has to be positive. The process involves isolating the variable and understanding that \( [x] \) is always an integer, which tells us that \( x \) must not be an integer to prevent division by zero. With such inequalities, you’re essentially dividing the number line into intervals and determining where the inequality holds true, hereby shaping the domain of the logarithmic function.

The integer part of a number, noted as \( [x] \), is a concept often found in more complex function domains. It represents the greatest integer that is less than or equal to \( x \). For example, if \( x = 3.7 \), then \( [x] = 3 \). The operation strips away any fractional part, leaving only the integer component.

Understanding the integer part function is pivotal because it has peculiar properties that influence inequalities and function domains. In our study case, the integer part function creates a 'step' nature in the number line, where every integer marks a potential discontinuity in the domain of \( f(x) \). Recognizing its effect helps you thoroughly analyze how the domain of the function is carved out.