Solve each equation. log￬x 27/64 = 3

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The equation log_b(a) = c means that b raised to the power of c equals a. Understanding this relationship is crucial for solving logarithmic equations, as it allows us to rewrite the logarithm in exponential form.

Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another, which can simplify calculations. It states that log_b(a) = log_k(a) / log_k(b) for any positive k. This is particularly useful when dealing with logarithms that are not easily computed in their original base.

Properties of Logarithms

Logarithms have several properties that simplify their manipulation, such as the product, quotient, and power rules. For example, log_b(mn) = log_b(m) + log_b(n) and log_b(m/n) = log_b(m) - log_b(n). These properties are essential for breaking down complex logarithmic expressions into simpler components.