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Logarithms come with a set of properties that are incredibly useful for simplifying and manipulating expressions. These properties can help break down complex problems into manageable steps. One important property is the product rule, which states that the logarithm of a product is the sum of the logarithms:

• Product Rule: \( \log_b (xy) = \log_b (x) + \log_b (y) \)

Another key property is the quotient rule, which helps when dealing with divisions:

• Quotient Rule: \( \log_b \left(\frac{x}{y}\right) = \log_b (x) - \log_b (y) \)

These rules make it easier to work with real-world applications by breaking down larger expressions into smaller parts. For this specific exercise, the most crucial property is the power rule, which we will discuss next.

The power rule is a fundamental property that plays a significant role in the expansion of logarithms. The power rule states:

• Power Rule: \( \log_b (x^y) = y \cdot \log_b (x) \)

This rule is particularly helpful when you have a logarithm of a number raised to a power. It allows you to bring the exponent down as a coefficient, simplifying the expression.
This is exactly what we did in the exercise with \( \ln \sqrt{a-1} \). By rewriting the square root as a power (\( (a-1)^{1/2} \)), we used the power rule to expand the expression to \( \frac{1}{2} \ln (a-1) \).
Utilizing these properties of logarithms, particularly the power rule, is efficient when simplifying complicated expressions.

Understanding how to handle square roots in mathematical expressions is an essential skill. When dealing with logarithms, it's helpful to recognize that a square root can be expressed as a fractional exponent. Specifically, for any number, \( x \), the square root is:

• Square Root: \( \sqrt{x} = x^{1/2} \)

Recognizing this property allows us to utilize the power rule of logarithms more effectively.
In our example, this property was crucial because it transformed \( \ln \sqrt{a-1} \) into \( \ln (a-1)^{1/2} \).
This allowed us to apply the power rule to simplify the expression further into \( \frac{1}{2} \ln (a-1) \). By converting square roots into exponents, you open the door to using logarithmic properties efficiently, making more complex calculations easier.