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Logarithmic functions are an essential part of algebra and offer a way to express exponential relationships. Simply put, a logarithm answers the question: "To what power must a base number be raised, to produce another number?" In the expression \( \log_{10}(x) \), the base is 10, which is common in scientific and financial calculations.

Logarithms follow a few key rules:

• The logarithm of 1 to any base is always 0, since any number raised to the power of 0 equals 1. Thus, \( \log_{10}(1) = 0 \).
• The logarithm of a base raised to its own power returns that power itself, so \( \log_{10}(10) = 1 \).
• They are undefined for negative numbers and zero, since no real number x exists such that 10 raised to it gives a negative number or zero.

Understanding these rules helps when examining the domain of functions involving logarithms. In the function given, \( f(x) = \left[ \log_{10}\left(\frac{5x-x^{2}}{4}\right) \right]^{1/2} \), the expression inside the logarithm needs to be positive and at least 1 to keep the logarithm non-negative, since \( \log_{10}(\text{something less than 1}) \) becomes negative. This ensures the function is well-defined over its domain.

Quadratic equations are fundamental in mathematics and appear in various forms of analysis and graphing. A standard quadratic equation is expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). In our exercise, we encountered rearranging into a quadratic form: \( x^2 - 5x + 4 \leq 0 \).

These equations are typically solved using:

• Factoring: If the quadratic can be factored into \( (px + q)(rx + s) = 0 \), we can find the roots \( x \) by setting each factor to zero.
• Quadratic Formula: When factoring is complex, apply the formula: \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \).
• Completing the Square: This transforms the quadratic into a perfect square trinomial, making it easier to handle.

For our specific quadratic \( x^2 - 5x + 4 = 0 \), we used the quadratic formula to find the roots: \( x = 3 \) and \( x = 1 \). These roots help us delineate intervals where the inequality \( x^2 - 5x + 4 \leq 0 \) holds true, which is crucial for determining the function's domain.

The domain of a function refers to all possible input values (usually \( x \)) which make the function work without any mathematical errors, such as dividing by zero or taking a logarithm of a negative number. In the function \( f(x) = \left[ \log_{10}\left(\frac{5x-x^{2}}{4}\right) \right]^{1/2} \), specifying the domain is critical as it involves a logarithm.

To define the domain here, we must ensure:

• The expression inside the logarithm \( \frac{5x-x^{2}}{4} \) is greater than or equal to 1, ensuring the logarithm is non-negative.
• The expression is always positive; otherwise, the logarithm itself becomes undefined.

Solving these conditions led us to form the quadratic inequality \( x^2 - 5x + 4 \leq 0 \), which indicates where the given expression remains valid. This inequality, after testing various intervals generated by its roots, showed that the function is well-defined for the interval \([1, 3]\). This knowledge allows us to understand for which \( x \) values the function can accurately be evaluated, ensuring it operates smoothly without mathematical inconsistencies.