Exponential Equations

Q: Why does matching the bases work?

Because with a fixed positive base (other than $1$), every exponent produces a different value — the outputs never repeat. So if $b^{x}$ and $b^{n}$ are equal, the only way that can happen is $x = n$.

Q: What about an equation like $4^{x} = 8$ where neither side is a power of the other?

Drop to a smaller shared base. $4 = 2^{2}$ and $8 = 2^{3}$, so the equation becomes $2^{2x} = 2^{3}$, giving $2x = 3$ and $x = \dfrac{3}{2}$. Fractional answers are normal here.

An exponential equation puts the variable up in the exponent: $4^{x} = 64$ or $3^{x+2} = 81$ . You can't subtract or divide your way to $x$ when it's upstairs — you need a different move.

The move is matching the bases. If two powers of the same base are equal, their exponents must be equal: $b^{x} = b^{n}$ means $x = n$ . Rewrite both sides as powers of one base, then the exponents give you an ordinary equation you already know how to solve.

The two-step plan

Step 1: rewrite both sides as powers of the same base. In $4^{x} = 64$ , notice $64 = 4^{3}$ , so the equation becomes $4^{x} = 4^{3}$ .

Step 2: set the exponents equal and solve. Same base on both sides means the exponents match: $x = 3$ .

This works because exponential expressions with a fixed base never repeat a value — each output comes from exactly one exponent. Equal outputs force equal exponents.

Getting good at spotting powers

The hard part is recognizing the target as a power. It helps to know the small power towers by heart: powers of $2$ ( $2, 4, 8, 16, 32, 64, 128$ ), powers of $3$ ( $3, 9, 27, 81, 243$ ), powers of $4$ ( $4, 16, 64, 256$ ), and powers of $5$ ( $5, 25, 125, 625$ ).

When the two sides use different bases, look for a smaller base underneath both. In $4^{x} = 8$ , neither is a power of the other, but both are powers of $2$ : $4 = 2^{2}$ and $8 = 2^{3}$ . Rewriting gives $2^{2x} = 2^{3}$ , so $2x = 3$ and $x = \dfrac{3}{2}$ .

When the exponent is an expression

If the exponent is something like $x + 2$ , nothing changes — the whole exponent gets set equal. $3^{x+2} = 81$ becomes $3^{x+2} = 3^{4}$ , so $x + 2 = 4$ and $x = 2$ . Matching bases turns every exponential equation into a one- or two-step equation.

Worked examples

Example 1: rewrite one side

Solve $2^{x} = 32$ .

Rewrite

32

as a power of

2

32 = 2^{5}

Match the bases

2^{x} = 2^{5}

Same base, so the exponents are equal

x = 5

Answer: $x = 5$

Example 2: an expression in the exponent

Solve $3^{x+2} = 81$ .

Rewrite

81

as a power of

3

81 = 3^{4}

Match the bases

3^{x+2} = 3^{4}

Set the exponents equal

x + 2 = 4

Solve

x = 2

Answer: $x = 2$

Example 3: rewrite both sides

Solve $4^{x} = 8$ .

Both sides are powers of

2

4 = 2^{2}, \quad 8 = 2^{3}

Rewrite the equation with base

2

2^{2x} = 2^{3}

Set the exponents equal

2x = 3

Solve

x = \dfrac{3}{2}

Answer: $x = \dfrac{3}{2}$

Try one yourself

Common questions

Why does matching the bases work?

Because with a fixed positive base (other than $1$ ), every exponent produces a different value — the outputs never repeat. So if $b^{x}$ and $b^{n}$ are equal, the only way that can happen is $x = n$ .

What if I can't write both sides with the same base?

Then the answer isn't a nice fraction and you'd need logarithms, which come later. In Algebra 1, the problems are built so a common base exists — if you're stuck, try the smallest base: can both sides be written as powers of $2$ or $3$ ?

What about an equation like

4^{x} = 8

where neither side is a power of the other?

Drop to a smaller shared base. $4 = 2^{2}$ and $8 = 2^{3}$ , so the equation becomes $2^{2x} = 2^{3}$ , giving $2x = 3$ and $x = \dfrac{3}{2}$ . Fractional answers are normal here.

How do I check my answer?

Substitute it back and evaluate. For $3^{x+2} = 81$ with $x = 2$ : $3^{4} = 81$ . ✓ If your check requires a calculator for a supposedly clean answer, recheck your power rewrite first.

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Exponential Equations

The two-step plan

Getting good at spotting powers

When the exponent is an expression

Worked examples

Example 1: rewrite one side

Example 2: an expression in the exponent

Example 3: rewrite both sides

Try one yourself

Common questions

Keep learning

Multiplication Rule

Power Rule

Power of a Fraction and Rational Exponents

Negative Exponent Rule