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The change of base formula is a critical logarithmic tool, allowing the expression of logarithms with an inconvenient base in terms of logarithms with a more suitable one. This technique not only provides flexibility in computations but also simplifies complex expressions. For example, if we have a logarithm with base \( a/b \) and wish to convert it to base \( a \), the formula \( \log_{b}a = \frac{\log_{c}a}{\log_{c}b} \) becomes vital. By selecting \( a \) as the new base, we rework \( \log_{a/b}x \) into \( \frac{\log_{a}x}{\log_{a}(a/b)} \) easing down the subsequent steps in solving logarithmic equations.

With changes of base, logarithms can be evaluated using a scientific calculator which often only has keys for \( \log \) (base 10) or \( \ln \) (base \( e \) - Euler's number). This accessibility makes the formula not just a theoretical construct but a practical armory in the toolkit of anyone dealing with logarithms.

The power rule for logarithms is an identity that enables us to handle exponents within logarithmic functions efficiently. It states that \( \log_{b}a^c = c \cdot \log_{b}a \). This rule permits the extraction of exponents, transforming multiplication operations into easier addition ones. In our step-by-step solution, when faced with \( \log_{a}(a/b) \) the power rule allows us to decompose this as \( \log_{a}a^1 - \log_{a}b \), which simplifies to \( 1 - \log_{a}b \).

Understanding the power rule is pivotal for algebraic manipulation involving logarithms. It helps in the simplification and resolution of equations where logarithms contain powers or roots, ensuring these operations do not pose an obstacle to learners eager to master logarithmic relations.

Logarithmic identities are the backbone of manipulating and simplifying logarithmic expressions. These identities are based on fundamental properties of logarithms that are universally true. For instance, \( \log_{b} b = 1 \) and \( \log_{b} 1 = 0 \). Another powerful identity used in the solution process is \( \log_{b} \frac{1}{a} = -\log_{b} a \). This identity reflects the inverse relationship between taking a logarithm of a number and its reciprocal.

By employing these identities, we see that \( - \log_{a}b \) directly translates into \( \log_{a}\frac{1}{b} \) completing the connection needed to equate the left and right sides of the equation in our exercise. Mastering logarithmic identities allows students to efficiently tackle a broad spectrum of problems involving logs, ensuring they are well-prepared for more advanced studies in mathematics.