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Logarithms have several useful properties that make complex expressions easier to evaluate. One of the most essential properties is the power rule. This rule states that \( \log_b{(x^a)} = a \cdot \log_b{x} \), where \( b \) is the logarithm base, \( x \) is the argument, and \( a \) is the exponent.Furthermore, the property that was used in the exercise is known as the identity property of logarithms. This states that \( a = \log_b{(b^a)} \). Essentially, this means the logarithm of a number where the base is the same number raised to some power is just the exponent itself.These properties allow us to simplify and solve logarithmic expressions without heavy calculations. They transform expressions in ways that make them more understandable and manageable.

Simplifying expressions can often mean rewriting them in different, yet equivalent forms. This makes calculations easier and ensures a better understanding of the relationship between the numbers involved.In our example, we started by simplifying \( \frac{1}{\sqrt{2}} \). Here, it's beneficial to recognize that \( \sqrt{2} \) can be expressed as a power of 2, specifically \( 2^{\frac{1}{2}} \). Consequently, its reciprocal becomes \( 2^{-\frac{1}{2}} \). This transformation is pivotal because it allows us to apply the properties of logarithms effectively.By simplifying expressions using exponents, we can directly apply logarithmic identities. It helps in avoiding complex fraction operations, which can sometimes be overwhelming.

Exponents are the foundation of many algebraic concepts, including logarithms. Understanding the rules governing exponents is crucial for effectively managing these calculations.There are several key rules to remember:

• Product Rule: \( x^a \times x^b = x^{a+b} \)
• Quotient Rule: \( \frac{x^a}{x^b} = x^{a-b} \)
• Power Rule: \( (x^a)^b = x^{a \cdot b} \)
• Negative Exponent Rule: \( x^{-a} = \frac{1}{x^a} \)

In the given exercise, the negative exponent rule plays a fundamental role. Transforming \( \frac{1}{\sqrt{2}} \) into \( 2^{-\frac{1}{2}} \) was a clear application of this rule, enabling further simplification and use of other logarithmic properties.