Equality and Inequality in Exponential Functions

Two powers with the same base can be compared without working out a single value. If they are equal, one short rule pins down the exponents. If one is larger, the answer depends entirely on whether the base is above 1 or between 0 and 1 — and in the second case the comparison runs backwards.

Concept The equality property

If two powers are equal and their bases are the same, then their exponents must be equal too. Nothing else could produce the same value.

Given

, the bases already match, so

This is the property that makes exponential equations solvable: rewrite both sides over a common base, then simply set the exponents equal to each other.

Case 1 Base greater than 1 — order is preserved

If and , then .

The function is increasing, so the values follow the exponents: a bigger exponent gives a bigger value, and a smaller exponent gives a smaller value.

With base 2: since

, it follows that

Checking directly,

Case 2 Base between 0 and 1 — order reverses

If and , then .

Here the function is decreasing, so the comparison flips: the larger exponent now produces the smaller value.

The reason is straightforward. Multiplying a fraction between 0 and 1 by itself makes it smaller, not larger:

Each extra factor shrinks the result further, so raising the exponent drives the value down.

Note The quick rule

Base above 1 — the order of the values matches the order of the exponents.

Base between 0 and 1 — the order of the values is the reverse of the order of the exponents.

Example Comparing two decaying powers

Which is larger, or ?

The base

lies between 0 and 1, so the function is decreasing.

That means the comparison reverses: the smaller exponent gives the larger value.

Since

, the power with exponent 3 wins.

Checking:

against

⟹ (1/2)³ is larger

Example Another decaying comparison

Compare and .

The base

is again between 0 and 1.

The exponents are 2 and 5, and

With a decaying base the smaller exponent gives the larger value.

⟹ (1/4)² > (1/4)⁵

Notice that neither example required evaluating the powers. Identifying the base and comparing the exponents settled it.

Example Using the equality property

Solve , and then for .

In the first, both sides share the base 2.

Equal powers with equal bases force equal exponents, so

In the second the base

is shared, so the exponents match.

⟹ x = 7, and y = 5

Note Mistakes to avoid

Applying the growth rule to a fractional base, and concluding that a bigger exponent always gives a bigger value.

Forgetting to check the base before comparing at all.

Using the equality property when the two bases are different — rewrite them over a common base first.

Assuming

is negative; it is small but still positive.

Evaluating both powers unnecessarily when the base and exponents already decide the answer.

Summary

If aˣ = aʸ with the same base, then x = y — this is what makes exponential equations solvable.
When a > 1 the function increases, so x > y gives aˣ > aʸ.
When 0 < a < 1 the function decreases, so x > y gives aˣ < aʸ — the comparison reverses.
The reversal happens because multiplying a fraction between 0 and 1 by itself makes it smaller.
Always identify the base first; it alone decides whether the order is preserved or reversed.