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Logarithmic properties are fundamental rules that simplify calculations involving logarithms. Understanding these properties can help in transforming complex expressions into simpler forms. The two log rules applied in the exercise are essential. The first one is the change of base formula, which rewrites a logarithm with an arbitrary base as a fraction involving common logarithms (base 10) or natural logarithms (base e). For example,

• \( \log_b a = \frac{\log a}{\log b} \)

Another important property is that the logarithm of a product can be expressed as the sum of logarithms:

• \( \log(ab) = \log a + \log b \)

These basic logarithmic properties allow us to handle logarithmic expressions and conduct reassessment of the problem effectively.

The base change formula is a powerful tool to transition between logarithms of different bases. This formula is crucial when dealing with logarithms, especially when calculators provide logarithms only in base 10 or base e. The base change formula states:

• \( \log_b a = \frac{\log_c a}{\log_c b} \)

Here, \( b \) is the base you want to convert from, and \( c \) is a convenient base like 10. This formula is used in the exercise to simplify the complex logarithmic function to base 10 logs, thereby making calculations possible using readily available tools such as a calculator.

Approximation is necessary when precise values are not easily attainable or practical to compute. In this exercise, the challenge is to find the value of the function to four decimal places. Approximation is achieved using calculated logarithms values, rounded to four decimal places:

• \( \log 5.3 \approx 0.7242 \)
• \( \log 4 \approx 0.6021 \)
• \( \log 8 \approx 0.9031 \)
• \( \log 1.2 \approx 0.0792 \)

These approximations allow calculations to be handled manually while remaining accurate enough to produce meaningful results. It is crucial to maintain precision by keeping calculations consistent to four decimal places at each step.

Following through complex expressions requires a systematic approach, often broken down into smaller, manageable steps. In the given problem, the calculation begins with substitution of known values into a simplified expression:1. Replace the variable \( x \) with its value.2. Simplify the components step by step using approximated logarithms values.3. Calculate individual terms first, such as finding \( \frac{\log 5.3}{\log 4} \) and other components separately.4. Subtract results as needed to solve the full expression accurately.Through these calculated steps, you can discern a final solution, which in this example came out to \( h(5.3) \approx -0.5764 \). Following logical calculation steps prevent errors and lead to a clear and reliable outcome.