Exponential equations ask when two exponential expressions are equal. Exponential inequalities ask when one is larger than the other — and the answer hinges on a single property of the base that changes everything about how the inequality behaves.
Key Principle: Base Determines Direction
The behavior of an exponential inequality depends entirely on whether the base is greater than 1 or between 0 and 1. This is the central idea on this page, and every solving method flows from it.
When a>1, the function ax is increasing — larger exponents produce larger values. So the inequality ax>ay holds exactly when x>y. The direction of the inequality is preserved.
When 0<a<1, the function ax is decreasing — larger exponents produce smaller values. So the inequality ax>ay holds exactly when x<y. The direction of the inequality flips.
The base a=1 is excluded because 1x=1 for all x — no inequality between distinct powers is possible.
This directional rule replaces the familiar "multiply or divide by a negative flips the inequality" from linear algebra. In exponential inequalities, it is not the sign of a multiplier but the size of the base that governs whether the inequality reverses.
Solving Basic Exponential Inequalities
The simplest exponential inequalities are solved by expressing both sides as powers of the same base and then applying the directional rule.
The inequality 2x>8 rewrites as 2x>23. Since the base 2 is greater than 1, the function is increasing and the inequality preserves direction: x>3.
The inequality (31)x>9 requires more care. Rewrite 9 as a power of 31: since 31−2=32=9, the inequality becomes (31)x>(31)−2. The base 31 is between 0 and 1, so the function is decreasing and the inequality flips: x<−2.
The inequality 5x−1≤125 rewrites as 5x−1≤53. Base greater than 1, direction preserved: x−1≤3, so x≤4.
The procedure is consistent: rewrite both sides with a common base, then read off the inequality between exponents — preserving direction if the base exceeds 1, reversing it if the base is a proper fraction.
Inequalities Requiring Simplification
When the two sides of an inequality do not immediately share a base, the laws of exponents are needed to rewrite one or both sides before comparison.
The inequality 4x<32 involves 4=22 and 32=25. Rewriting: (22)x<25, so 22x<25. Base 2>1, direction preserved: 2x<5, giving x<25.
The inequality 9x+1≥27x involves 9=32 and 27=33. Rewriting: 32(x+1)≥33x, which gives 32x+2≥33x. Base 3>1, direction preserved: 2x+2≥3x, so 2≥x, meaning x≤2.
The inequality (41)x>(81)2 requires converting both bases. Since 41=2−2 and 81=2−3, the inequality becomes (2−2)x>(2−3)2, or 2−2x>2−6. Base 2>1: −2x>−6, so x<3.
The algebraic manipulation happens before the directional rule is applied. Simplify first, compare second.
Domain and Sign Considerations
Exponential expressions with positive bases carry a property that constrains the solution space: ax>0 for every real x when a>0.
No real exponent can make a positive base produce zero or a negative result. The equation 2x=0 has no solution. The inequality 3x<0 has no solution. This fact is not just a technicality — it eliminates entire branches of potential answers.
The inequality 2x>−5 is satisfied by every real x, because 2x is always positive and thus always greater than −5. No computation is needed once the sign property is recognized.
The inequality 2x<−1 has no solution at all, for the same reason.
When negative exponents appear with variable bases, domain restrictions must be checked. The expression x−2>4 requires x=0, and the solution set must exclude zero regardless of what the algebra produces. Similarly, expressions involving rational exponents with even roots require the base to be non-negative.
Compound Inequalities
A compound exponential inequality places an exponential expression between two bounds, requiring the variable to satisfy both constraints simultaneously.
The inequality 41<2x<16 sets lower and upper bounds on 2x. Rewrite each bound as a power of 2: 2−2<2x<24. Since the base 2>1 preserves direction, the solution is −2<x<4.
The inequality 271≤3x−1≤81 rewrites as 3−3≤3x−1≤34. Preserving direction: −3≤x−1≤4, so −2≤x≤5.
With a base between 0 and 1, both inequality directions flip. The inequality 91<(31)x<3 rewrites as (31)−2<(31)x<(31)−1. Since 31<1, the function is decreasing: −2>x>−1, which reads as −1<x<−2 — but this is empty when written carelessly. Reversing properly: −2<x<−1. Care with the direction at each step prevents this kind of error.
Systems involving multiple exponential inequalities with different bases are handled by solving each inequality independently and then intersecting the solution sets.