Solving Exponential and Logarithmic Equations

Exponential and logarithmic equations are inverse forms, so understanding their relationship helps you solve many problems. An exponential equation has the variable in the exponent, such as 3^x = 81, and you can solve it by rewriting both sides with the same base or by using logarithms. When both sides share a common base, equate the exponents directly, because exponential functions with the same base are one-to-one. If rewriting to a common base is impractical, apply a logarithm to both sides, using a natural log or common log to bring the exponent down as a coefficient. The properties of logarithms — product, quotient, and power rules — let you combine or expand terms to isolate the variable conveniently. For example, solving 2^x = 7 requires taking log of both sides, then x = log(7)/log(2) by the change of base formula. For logarithmic equations, first use properties to condense multiple logs into a single expression and then exponentiate both sides to remove the log. Always check your solutions because domain restrictions, such as positive arguments for logs, can make apparent answers invalid. Practice choosing whether to convert bases, apply log rules, or use change of base to streamline each problem. With these strategies, you can tackle a wide range of exponential and logarithmic equations encountered in algebra and precalculus.