Solve each equation. Give solutions in exact form. See Examples 5–9. log_6 (2x + 4) = 2

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 (b^c = a). Understanding this relationship is crucial for solving logarithmic equations, as it allows us to rewrite the logarithmic expression in exponential form.

Properties of Logarithms

Properties of logarithms, such as the product, quotient, and power rules, help simplify logarithmic expressions. For instance, 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 manipulating equations involving logarithms to isolate the variable.

Solving Exponential Equations

To solve an equation involving logarithms, we often convert it into an exponential equation. For example, from log_6(2x + 4) = 2, we can rewrite it as 6^2 = 2x + 4. This transformation allows us to solve for the variable by isolating it, which is a fundamental skill in algebra.