91Ó°ÊÓ

Logarithmic functions are powerful mathematical tools used to solve various types of equations, especially those involving exponential growth and decay. In this exercise, the function under consideration is given by \( f(x) = \ln \left(1 - \frac{1}{x^2} \right) \). Here, \( \ln \) represents the natural logarithm, which is logarithm with the base \( e \), a mathematical constant approximately equal to 2.718.
To simplify expressions like this, it's crucial to utilize the properties of logarithms. One fundamental property is that \( \ln a + \ln b = \ln(ab) \). In simple terms, this means that the logarithm of a product can be split into the sum of the logarithms. In the given exercise, this property allows us to combine the separate logarithmic terms, \( f(2), f(3), \) and \( f(4) \) into a single expression of \( \ln \left(\frac{5}{8}\right) \).
By understanding and applying these properties, solving logarithmic equations becomes more straightforward and less intimidating.

Fraction multiplication is an essential arithmetic operation used frequently when simplifying various mathematical expressions, including those involving logarithms. Here, we need to multiply several fractions as part of the simplification process in the problem.
Working through our example, consider the fractions involved: \( \frac{3}{4} \), \( \frac{8}{9} \), and \( \frac{15}{16} \). Multiplying fractions involves multiplying the numerators together and the denominators together:

• For the numerators: \( 3 \times 8 \times 15 = 360 \).
• For the denominators: \( 4 \times 9 \times 16 = 576 \).

As a result, we have the fraction \( \frac{360}{576} \). It's important to simplify this resulting fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor, which is 72 in this case, resulting in \( \frac{5}{8} \). Perfectly executing fraction multiplication and simplification ensures a correct and simplified logarithmic result.

Simplifying mathematical expressions involves reducing them into their simplest forms, making them easier to work with and understand. In the context of the given problem, simplification is achieved through a series of steps utilizing the properties of logarithms and fraction multiplication.
The initial expressions \( 1 - \frac{1}{4} \), \( 1 - \frac{1}{9} \), and \( 1 - \frac{1}{16} \) can be directly evaluated to produce simpler fractions: \( \frac{3}{4} \), \( \frac{8}{9} \), and \( \frac{15}{16} \), respectively. Each expression in this stage is simplified by executing basic subtraction and finding a common fraction form.
Subsequently, when we have to handle the fraction multiplication that follows: \( \frac{3}{4} \times \frac{8}{9} \times \frac{15}{16} \), the role of simplification is emphasized again. The process culminates as these values are worked through, showing that the multiplication of simplified fractions aligns with logarithmic combination rules, leading to \( \ln \left(\frac{5}{8} \right) \). Mastery of simplification techniques is vital for solving complex problems efficiently, as it reduces the complexity and potential for error while calculating.