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The natural logarithm is a logarithmic function that is denoted as \( \ln(x) \), and it is fundamentally related to the constant \( e \), which is approximately equal to 2.71828. This special logarithm is so named because its base is \( e \), and it arises naturally in many areas of mathematics, particularly in problems dealing with growth and decay.

For example, if you have an amount that is growing continuously at a certain rate, the time it takes for the amount to grow to a specific value can be represented using the natural logarithm. It is also a key element in calculus, appearing in integrals and differential equations. In the context of expanding logarithmic expressions, understanding the natural logarithm is crucial, as it leads to simplifications such as \( \ln e = 1 \) because \( e \) is the base of natural logarithms. This property is exploited to simplify expressions as seen in the provided exercise.

The power rule of logarithms states that the logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the number itself. Mathematically, it is expressed as \( \ln a^m = m \ln a \). This property is incredibly useful for simplifying complex logarithmic expressions, making them more manageable.

When dealing with square roots, remember that a square root is equivalent to raising a number to the \( \frac{1}{2} \) power. This is why in the exercise's step 1, the expression within the square root, \( \sqrt{ex} \), can be rewritten as \( (ex)^{\frac{1}{2}} \) leveraging the power rule. This transformation lays the foundation for expanding and simplifying the overall logarithmic expression.

The product rule of logarithms is a fundamental property that helps break down logarithms of products into more digestible parts. This rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors: \( \ln(ab) = \ln a + \ln b \).

By applying this rule to logarithmic expressions involving products, as seen in step 3 of the exercise, you can simplify the expression by writing it as a sum of separate logarithms. For instance, this rule transforms \( \ln(ex) \) into \( \ln e + \ln x \) which makes the expression easier to manage and opens up opportunities for further simplification like evaluating \( \ln e \) as 1, since the base of the natural logarithm is \( e \).

Working with logarithmic expressions requires an understanding of various logarithmic properties and how to apply them correctly. These expressions can often be transformed and simplified using rules such as the power rule for exponents within a logarithm and the product rule for products inside a logarithm.

As you approach an expression like \( \ln \sqrt{ex} \), identifying and applying these properties systematically can expand the expression into a simpler or more useful form. It's much like untangling a knot—each logarithmic property you apply helps to pull the correct threads, leading you to a clearer understanding of the resulting expression. The final simplified form, such as \( \frac{1}{2} + \frac{1}{2} \ln x \) obtained in the exercise, is more than just an answer—it's a demonstration of how the properties of logarithms interact and flow together.