Exponential Equations and Inequalities
A rigorous, technique-driven guide — from same-base reduction to substitution, homogeneous forms, and inequality analysis.
1. Prerequisites Recap
1.1 Laws of Exponents
Every technique in this lesson rests on three fundamental laws. For any base $a > 0$ and real exponents $x, y$:
| Law | Statement | Quick Example |
|---|---|---|
| Product Rule | $a^x \cdot a^y = a^{x+y}$ | $2^3 \cdot 2^5 = 2^{8} = 256$ |
| Power Rule | $(a^x)^y = a^{xy}$ | $(3^2)^4 = 3^{8} = 6561$ |
| Negative Exponent | $a^{-x} = \dfrac{1}{a^x}$ | $5^{-2} = \dfrac{1}{25}$ |
| Quotient Rule | $\dfrac{a^x}{a^y} = a^{x-y}$ | $\dfrac{7^5}{7^2} = 7^3 = 343$ |
1.2 The Exponential Function $f(x) = a^x$
- Domain: $(-\infty, \infty)$ — every real number is a valid exponent.
- Range: $(0, \infty)$ — the output is always positive. This is crucial: $a^x > 0$ for all $x$.
- Monotonicity:
- If $a > 1$: $f$ is strictly increasing — larger exponents give larger values.
- If $0 < a < 1$: $f$ is strictly decreasing — larger exponents give smaller values.
Why monotonicity matters: A strictly monotonic function is one-to-one (injective). This gives us the One-to-One Property:
$$a^{f(x)} = a^{g(x)} \iff f(x) = g(x) \qquad (a > 0,\; a \neq 1)$$
In words: if the bases are equal and the exponential values are equal, then the exponents must be equal. This single fact is the engine behind most of our solution methods.
Interactive: Explore $y = a^x$
Drag the slider to change the base $a$ and observe how the graph changes between increasing ($a>1$) and decreasing ($0
Interactive visualization — available in the full lesson
2. Basic Exponential Equations — Same-Base Reduction
The simplest exponential equations are those where both sides can be rewritten with the same base.
Strategy: Rewrite every term as a power of a common base $a$, then apply the one-to-one property:
$$a^{f(x)} = a^{g(x)} \implies f(x) = g(x)$$