Retaining the Concepts. Expand: log7 (5√x/49y^10) fifth root of x

Logarithmic properties, such as the product, quotient, and power rules, are essential for simplifying logarithmic expressions. The product rule states that log_b(mn) = log_b(m) + log_b(n), the quotient rule states log_b(m/n) = log_b(m) - log_b(n), and the power rule states log_b(m^k) = k * log_b(m). Understanding these rules allows for the effective manipulation of logarithmic expressions.

Radical expressions involve roots, such as square roots or cube roots, and can be expressed in exponential form. For example, the fifth root of a number can be represented as raising that number to the power of 1/5. Recognizing how to convert between radical and exponential forms is crucial for simplifying expressions that include roots.

The change of base formula allows for the conversion of logarithms from one base to another, which is particularly useful when dealing with logarithms that are not easily simplified. The formula states that log_b(a) = log_k(a) / log_k(b) for any positive k. This concept is important for evaluating logarithmic expressions when the base is not standard or when simplification is needed.