Exponents, Logarithms and their Applications — Exponential equations

About this lesson

This unit focuses on solving exponential equations, both algebraically and graphically. It begins by defining an exponential equation as one where the unknown appears in the exponent, such as 3^x = 27. The unit presents two main algebraic rules: first, if a^m = a^n (with a not equal to 0, 1, or -1), then m = n; second, if a^m = b^m (with a and b not equal to 0, 1, or -1), then either m = 0 or a = b (if m is odd) or a = ±b (if m is even). Worked examples show how to apply these rules to find solution sets in R. The unit also demonstrates solving exponential equations graphically by plotting two functions (e.g., f(x) = 2^x and g(x) = 3 - x) and finding their intersection point. Additionally, it includes problems involving function notation, such as proving identities like f(x+2) × f(x-2) = f(2x) for f(x) = 3^x, and solving equations that combine exponential terms, sometimes requiring substitution or factoring. The unit ends with a set of exercises covering various types of exponential equations, including those with different bases, and a creative thinking problem. By the end, a student should be able to solve exponential equations algebraically using the stated rules, solve them graphically using technology, and apply these skills to problems involving functions and real-world patterns.