Exponential Equations and Inequalities

Properties of Exponents

In an exponential equation, the unknown is in the exponent: aˣ = b (a > 0, a ≠ 1).

Property	Formula
Product	aᵐ · aⁿ = aᵐ⁺ⁿ
Quotient	aᵐ / aⁿ = aᵐ⁻ⁿ
Power of a power	(aᵐ)ⁿ = aᵐⁿ
Negative exponent	a⁻ⁿ = 1 / aⁿ
Fractional exponent	a^(m/n) = ⁿ√(aᵐ)

aˣ is always positive: aˣ > 0 for all x, so aˣ = −3 has no solution.

The Simplest Equations and Inequalities

• If aᶠ = aᵍ, then f = g (a > 0, a ≠ 1).
• If a > 1, the inequality aᶠ > aᵍ is equivalent to f > g: the sign is kept.
• If 0 < a < 1, the inequality aᶠ > aᵍ is equivalent to f < g: the sign is reversed.

Solution Methods

1. Reduce both sides to the same base
2. If the bases are equal, equate the exponents
3. If it is complicated, take out the common factor or substitute t = aˣ
4. Do not forget the condition t > 0
5. Substitute back and find x

• Common factor: 3ˣ⁺¹ + 3ˣ = 3ˣ · (3 + 1) = 4 · 3ˣ.
• Quadratic substitution: 4ˣ = (2ˣ)², 9ˣ = (3ˣ)².
• In the substitution t = aˣ, t > 0. The root with negative t is discarded.
• Case of different bases (of the form 2ˣ = 3ˣ): divide both sides by 3ˣ to get (2/3)ˣ = 1, x = 0.

Examples

• Since 2ˣ⁻¹ = 16 = 2⁴, we get x − 1 = 4, x = 5.
• 4ˣ − 5 · 2ˣ + 4 = 0. Let t = 2ˣ: t² − 5t + 4 = 0, t = 1 or t = 4. So x = 0 or x = 2.
• (1/3)ˣ > 9. This is (1/3)ˣ > (1/3)⁻², the base is less than 1, so x < −2.