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The quotient rule of logarithms is a powerful tool that simplifies logarithmic expressions involving subtraction. This rule states that for the same base, the difference of two logarithms is equivalent to a single logarithm with a quotient of the arguments. Mathematically, it is expressed as: \[ \log_{b}(A) - \log_{b}(B) = \log_{b}\left(\frac{A}{B}\right) \]In the original exercise, the equation \( \log_{5}(x+1) - \log_{5}(x-1) = 2 \) applied the quotient rule. This combined the two separate logarithms into one:

• \( \log_{5}\left(\frac{x+1}{x-1}\right) = 2 \)

This step significantly simplifies the problem, reducing the complexity of further calculations. Understanding the quotient rule is crucial when dealing with more advanced logarithmic problems, as it helps streamline the equation into a more manageable form.

Converting a logarithmic equation to its exponential form is essential in solving equations involving logarithms. Remember, a logarithmic expression \( \log_{b}(A) = C \) can be transformed into an exponential form as \( A = b^{C} \). In the given problem, the equation transformed into exponential form is:

• \( \log_{5}\left(\frac{x+1}{x-1}\right) = 2 \) becomes \( \frac{x+1}{x-1} = 5^{2} \)

This conversion is important because it translates the problem from logarithmic language, which can sometimes be unintuitive, into a format that is often easier to manipulate algebraically. By recognizing this equivalence, you can simplify and continue working towards solving for \( x \). It's an accessible way to approach a logarithmic problem using familiar exponential functions.

After converting the logarithmic expression to an exponential form, you encounter a proportion, a mathematical equation that states two ratios or fractions are equivalent. In this scenario, \( \frac{x+1}{x-1} = 25 \).To solve this proportion, cross-multiplication is used, which involves multiplying the numerator of one fraction by the denominator of the other and vice-versa. The steps are as follows:

• Cross-multiply: \( x+1 = 25(x-1) \)
• Distribute the 25: \( x+1 = 25x - 25 \)

This prepares the equation for solving by isolating \( x \). Cross-multiplying is a reliable method to solve proportions, allowing for straightforward simplification of the equation. Understanding proportions will enhance your algebra skills, giving you tools to deal with a wide range of mathematical problems involving fractions or ratios.