Exponential equations and inequalities

Fundamental principle

As the exp function is strictly increasing on R (and therefore injective), for all real numbers a and b:

e^a = e^b <=> a = b

e^a < e^b <=> a < b

This allows us to solve equations and inequalities by directly comparing the exponents.

Example equation

Let’s solve e^(2x-1) = e^(x+3):

e^(2x-1) = e^(x+3) ⇔ 2x - 1 = x + 3 ⇔ x = 4

The solution set is S = {4}.

Example with e^x = k

Let’s solve e^x = 5. As 5 > 0, a solution exists: x = ln(5) (the natural logarithm, the reciprocal function of e, will be covered later). If the equation were e^x = -3, there would be no solution because e^x is always greater than 0.

Example of an inequality

Let’s solve e^(3x) ≤ e^(x+4):

e^(3x) ≤ e^(x+4) ⇔ 3x ≤ x + 4 ⇔ 2x ≤ 4 ⇔ x ≤ 2

The solution set is S = ]–∞; 2].

Common pitfall

Never divide or multiply an inequality by e^x without considering the sign: in fact, e^x is always strictly positive, so you can multiply by e^x without ever changing the direction of the inequality. This is an advantage to be used wisely, but you must justify it.

Key points

• e^a = e^b implies a = b.
• e^a < e^b implies a < b.
• An equation e^x = k where k ≤ 0 never has a solution.