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When dealing with logarithmic expressions, the power rule is a handy tool to simplify and solve problems more efficiently. The power rule states that you can bring the exponent of the inside term to the front as a coefficient. Mathematically, it’s written as:\[ p \log_b(x) = \log_b(x^p) \]This is particularly useful when working with expressions that contain coefficients in front of the log term. Instead of calculating directly, you first raise the base of the logarithm to the power of the coefficient.

• Example: If you encounter an expression like \(3 \log_b(2)\), you apply the power rule to rewrite it as \(\log_b(2^3)\) which is \(\log_b(8)\).
• This helps in simplifying further operations, such as combining the logs in a later step.

By using the power rule, each term in our original expression can be rewritten without extra coefficients affecting the simple single-log form. This forms the foundation for combining logarithms effectively in subsequent steps.

The quotient rule for logarithms simplifies expressions where logs are being subtracted. It states:\[ \log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right) \]This rule allows you to write a single logarithm expression by transforming the subtraction of two logs into a division inside one log. This can be particularly useful when you have multiple log terms to consolidate.

• If initially, you have \(\log_b(15) - \log_b(3)\), then applying the quotient rule, you have \(\log_b\left(\frac{15}{3}\right)\) which simplifies to \(\log_b(5)\).

Understanding this rule is crucial when working with problems that involve both multiplication and division of the arguments within logarithms. It streamlines the process of combining them into a neater form, which is easier to manage.

Combining logarithmic terms into a single expression is an essential skill for simplifying complex equations. This involves using both the power and quotient rules for logarithms.After applying the power rule to eliminate coefficients, the next step is usually merging the logs into one. Once you have similar base logs, use both the product and quotient rules accordingly:

• First, apply the product rule: If adding, you multiply the numbers inside the logs.
• Then, apply the quotient rule: If subtracting, you divide the numbers inside the logs.

In our example, we rearranged terms to apply these principles. The original separate logarithms of \(p\log_b(A)\), \(q\log_b(B)\), and \(r\log_b(C)\) are combined by rearranging into one single log:\[ \log_b\left( \frac{A^p \cdot C^r}{B^q} \right) \]This result is a single logarithm, elegantly capturing the full essence of the equation, with no coefficients, fulfilling the requirement for a coefficient of one. It is a powerful example of using logarithm properties to streamline complex expressions into simplified forms.