How do you solve exponential equations by matching bases?

Rewrite both sides as powers of the same base, then set exponents equal. For 2ˣ = 8: since 8 = 2³, the equation becomes 2ˣ = 2³, so x = 3. This works because aˣ = aʸ implies x = y when a > 0 and a ≠ 1.

How do you solve 4^x = 8?

Convert both to powers of 2: 4 = 2² and 8 = 2³. The equation becomes (2²)ˣ = 2³, which simplifies to 2^(2x) = 2³. Match exponents: 2x = 3, so x = 3/2.

What if the bases cannot be matched?

When no common base exists (like 2ˣ = 5), logarithms are required. Take log of both sides: x·log(2) = log(5), so x = log(5)/log(2). This topic is covered in the logarithms section.

How do you solve equations with rational exponents like x^(2/3) = 4?

Raise both sides to the reciprocal power. For x^(2/3) = 4: raise to power 3/2 to get x = 4^(3/2) = (√4)³ = 2³ = 8. The reciprocal exponent undoes the original.

How do you solve x^(-2) = 9?

Rewrite as 1/x² = 9, so x² = 1/9, giving x = ±1/3. Negative exponents mean reciprocals, so isolate the positive exponent first, then solve normally.

How do substitution and quadratic methods work for exponential equations?

When an equation has a^(2x) and a^x terms, substitute t = aˣ. For 4ˣ - 3·2ˣ + 2 = 0: since 4ˣ = (2ˣ)², let t = 2ˣ. Get t² - 3t + 2 = 0, factor to (t-1)(t-2) = 0, giving t = 1 or t = 2, so x = 0 or x = 1.

Why do exponential equations sometimes have no solution?

Because aˣ > 0 for any positive base and real exponent. The equation 2ˣ = -4 has no solution since 2ˣ is always positive. Similarly, when substitution gives a negative value for t = aˣ, that root must be rejected.

What are extraneous solutions in exponential equations?

Solutions that emerge from algebra but don't satisfy the original equation. Substitution can produce negative t values (invalid since aˣ > 0). Squaring can introduce false roots. Always substitute answers back into the original equation to verify.

What's the difference between exponential and polynomial equations?

In exponential equations (2ˣ = 8), the variable is in the exponent. In polynomial equations (x³ = 8), the variable is the base. They require completely different solving techniques: matching bases vs taking roots or factoring.

Exponential Equations

Q: What is an exponential equation?

An exponential equation has the variable in the exponent, like 2ˣ = 16 or 3^(2x-1) = 27. This differs from polynomial equations like x³ = 8 where the variable is the base. The position of the variable determines which solving techniques apply.

What is an Exponential Equation

Forms of Exponential Equations

Preparing the Equation

Solving Methods Overview

Method: Matching Bases

Method: Using Exponent Laws

Method: Using Logarithms

Method: Substitution (Quadratic in Disguise)

Equations with Rational and Negative Exponents

Checking Solutions

Solving for the Exponent

In a polynomial equation like

x^2 = 9

, the unknown is the base and the exponent is fixed. In an exponential equation, the roles reverse — the base is known and the unknown sits in the exponent. The equation

2^x = 8

asks not "what number squared gives

9

?" but "how many times must

2

be multiplied by itself to reach

8

?" That shift demands a different set of solving techniques, all grounded in the laws of exponents.

What is an Exponential Equation

An exponential equation is any equation in which the variable appears in an exponent. The equation

2^x = 16

is exponential — the unknown

x

controls how many times the base is applied. The equation

3^{2x-1} = 27

is exponential. The equation

5 \cdot 4^x = 320

is exponential.

What makes these equations distinct from polynomial equations is the position of the variable. In

x^3 = 8

, the variable is the base and the exponent is a fixed number — this is a polynomial equation, solved by taking a root. In

2^x = 8

, the base is fixed and the exponent is the variable — this is an exponential equation, and roots alone will not solve it.

The distinction matters because the techniques are entirely different. Polynomial equations are solved by factoring, applying the quadratic formula, or extracting roots. Exponential equations require rewriting expressions using the laws of exponents, matching bases, or — when those approaches fail — logarithms.

Forms of Exponential Equations

Exponential equations appear in several recognizable forms, and identifying the form determines which solving method applies.

A single exponential equal to a constant —

2^x = 16

— is the simplest case. If the constant is a power of the same base, matching bases resolves it immediately.

A coefficient multiplying the exponential term —

3 \cdot 2^x = 24

— requires isolating the exponential first. Divide both sides by

3

to get

2^x = 8

, then proceed.

A linear expression in the exponent —

2^{3x+1} = 32

— adds one algebraic step after the bases are matched:

3x + 1 = 5

, so

x = \frac{4}{3}

.

An additive constant alongside the exponential —

2^x + 5 = 13

— must be moved before anything else. Subtract

5

to isolate

2^x = 8

.

Both sides exponential with convertible bases —

4^x = 8

— calls for rewriting to a common base using the power of a power rule.

Both sides exponential with genuinely different bases —

2^x = 3^{x-1}

— cannot be solved by matching bases. These require logarithms.

A sum of same-base terms —

2^x + 2^{x+1} = 12

— needs factoring via the product rule before the exponential can be isolated.

A quadratic structure in disguise —

4^x - 3 \cdot 2^x + 2 = 0

— hides a polynomial equation under the exponential notation, solved by substitution.

Preparing the Equation

Before applying any method, the equation must be arranged so that the exponential structure is exposed.

The first priority is isolating the exponential term. Move additive constants to the opposite side:

5 \cdot 3^x + 7 = 252

becomes

5 \cdot 3^x = 245

. Divide out multiplicative coefficients:

3^x = 49

.

The second priority is recognizing common bases. The numbers

2, 4, 8, 16, 32, 64

are all powers of

2

. The numbers

3, 9, 27, 81

are powers of

3

. The numbers

5, 25, 125

are powers of

5

. Spotting these relationships determines whether matching bases is feasible.

The third priority is rewriting bases before solving. The equation

8^x = 4^{x+3}

looks like it has different bases, but

8 = 2^3

and

4 = 2^2

. Applying the power of a power rule gives

2^{3x} = 2^{2(x+3)}

, and the equation is ready for base matching.

The fourth priority is spotting the quadratic pattern. If the equation contains both

a^{2x}

and

a^x

— recognizable because

a^{2x} = (a^x)^2

by the power of a power rule — substitution will convert it into a polynomial equation.

Preparation is not a separate step from solving — it is the step that makes solving possible.

Solving Methods Overview

Four methods cover virtually all exponential equations encountered in algebra.

Matching bases rewrites both sides as powers of the same base, then equates the exponents. It works when the bases are equal or convertible to a common base through the laws of exponents.

Using exponent laws restructures the equation by factoring, applying the product rule, or combining terms. It works when the equation contains multiple exponential terms with the same base that need to be consolidated before the exponential can be isolated.

Using logarithms takes the log of both sides and brings the exponent down using the identity

\log(a^x) = x \cdot \log(a)

. It works for any exponential equation, but is essential when the bases on both sides are genuinely different and no common base exists.

Substitution replaces

a^x

with a temporary variable

t

, converting the exponential equation into a polynomial — typically a quadratic. It works when the equation involves both

a^{2x}

and

a^x

, creating a squared-and-linear pattern that factors cleanly.

Each method has its natural domain. Matching bases is fastest when it applies. Exponent laws handle multi-term equations. Logarithms are the universal fallback. Substitution targets disguised polynomials.

Method: Matching Bases

The simplest exponential equations yield to a single principle: if two powers of the same base are equal and the base is positive and not equal to

1

, then their exponents must be equal.

a^x = a^y \implies x = y \qquad (a > 0,\; a \neq 1)

This works because exponential functions with base

a > 0

a \neq 1

are one-to-one — different exponents always produce different outputs.

For

2^x = 8

, recognize that

8 = 2^3

. The equation becomes

2^x = 2^3

, so

x = 3

.

The method extends to cases where the connection between bases is less obvious. The equation

4^x = 8

involves different bases, but both are powers of $2$:

4 = 2^2

and

8 = 2^3

. Rewriting gives

(2^2)^x = 2^3

, which by the power of a power rule becomes

2^{2x} = 2^3

. Matching exponents:

2x = 3

, so

x = \frac{3}{2}

.

The equation

9^{x+1} = 27

works the same way. Both bases are powers of

3

9 = 3^2

and

27 = 3^3

. Rewriting:

(3^2)^{x+1} = 3^3

, giving

3^{2(x+1)} = 3^3

. So

2(x+1) = 3

, and

x = \frac{1}{2}

.

The key skill is recognizing when two numbers share a common base. Familiarity with small powers —

2, 4, 8, 16, 32

;

3, 9, 27, 81

;

5, 25, 125

— makes this recognition faster.

Method: Using Exponent Laws

Some exponential equations require algebraic manipulation before the bases can be matched. The laws of exponents provide the tools for restructuring these expressions.

Consider

2^{x+1} + 2^x = 12

. The term

2^{x+1}

can be rewritten using the product rule:

2^{x+1} = 2^x \cdot 2^1 = 2 \cdot 2^x

. The equation becomes

2 \cdot 2^x + 2^x = 12

.

Now

2^x

is a common factor:

2^x(2 + 1) = 12

, which simplifies to

3 \cdot 2^x = 12

, so

2^x = 4 = 2^2

, giving

x = 2

.

Consider

3^{x+2} - 3^x = 72

. Rewrite

3^{x+2} = 3^x \cdot 3^2 = 9 \cdot 3^x

. The equation becomes

9 \cdot 3^x - 3^x = 72

, then

3^x(9 - 1) = 72

, so

8 \cdot 3^x = 72

and

3^x = 9 = 3^2

. Thus

x = 2

.

The general strategy is to express every exponential term as a product involving a single exponential expression — the same technique used to factor polynomials by pulling out a common term. Factor out that common piece, reduce the equation to a single exponential term equal to a constant, and finish with base matching or logarithms.

Method: Using Logarithms

When the bases on both sides of an equation cannot be matched — because no common base exists — logarithms provide the way forward.

The core identity is

\log(a^x) = x \cdot \log(a)

. Taking the logarithm of both sides brings the exponent down from its position as a power to the position of a multiplier, where it can be isolated algebraically.

For

2^x = 5

, no integer or simple rational power of

2

equals

5

. Take the logarithm of both sides:

x \cdot \log(2) = \log(5)

. Divide:

x = \frac{\log(5)}{\log(2)} \approx 2.322

.

For

3^{x+1} = 7

, take logarithms:

(x+1) \cdot \log(3) = \log(7)

. Then

x + 1 = \frac{\log(7)}{\log(3)}

and

x = \frac{\log(7)}{\log(3)} - 1 \approx 0.771

.

For

2^x = 3^{x-1}

, both sides are exponential with different bases. Take logarithms:

x \cdot \log(2) = (x - 1) \cdot \log(3)

. Distribute:

x \cdot \log(2) = x \cdot \log(3) - \log(3)

. Collect

x

x(\log(2) - \log(3)) = -\log(3)

. Solve:

x = \frac{\log(3)}{\log(3) - \log(2)} \approx 2.710

.

Logarithms work universally — they solve any exponential equation, including those where matching bases also works. The other methods are shortcuts; logarithms are the general tool.

Method: Substitution (Quadratic in Disguise)

Some exponential equations disguise a familiar structure. The equation

4^x - 3 \cdot 2^x + 2 = 0

looks intimidating, but a substitution reveals a quadratic hiding underneath.

Since

4^x = (2^2)^x = (2^x)^2

by the power of a power rule, let

t = 2^x

. The equation becomes:

t^2 - 3t + 2 = 0

This factors as

(t - 1)(t - 2) = 0

, giving

t = 1

t = 2

.

Substituting back:

2^x = 1

gives

x = 0

(since any positive base raised to zero equals $1$), and

2^x = 2

gives

x = 1

. Both solutions are valid.

The substitution works whenever the equation involves two exponential terms where one exponent is double the other —

a^{2x}

and

a^x

, or

a^{4x}

and

a^{2x}

. The doubled exponent creates a perfect square under substitution, converting the exponential equation into a polynomial one.

More complex examples follow the same pattern. The equation

9^x + 2 \cdot 3^x - 15 = 0

becomes

t^2 + 2t - 15 = 0

with

t = 3^x

, factoring as

(t + 5)(t - 3) = 0

. Since

3^x > 0

for all real

x

(the exponential function is always positive), the solution

t = -5

is rejected, leaving

3^x = 3

and

x = 1

Equations with Rational and Negative Exponents

Not every equation with exponents has the variable in the exponent. Some place the variable in the base while the exponent is a fixed rational or negative number. These are solved by inverting the exponent.

The equation

x^{2/3} = 4

asks: what value of

x

, when raised to the power

\frac{2}{3}

, gives

4

? To isolate

x

, raise both sides to the reciprocal power

\frac{3}{2}

x = 4^{3/2} = (\sqrt{4})^3 = 2^3 = 8

The equation

x^{-2} = 9

translates to

\frac{1}{x^2} = 9

using the negative exponent definition, so

x^2 = \frac{1}{9}

, giving

x = \pm\frac{1}{3}

.

The equation

x^{3/4} = 27

is solved by raising both sides to the power

\frac{4}{3}

x = 27^{4/3} = (\sqrt[3]{27})^4 = 3^4 = 81

.

The reciprocal exponent works because

(a^{m/n})^{n/m} = a^1 = a

by the power of a power rule. The exponent and its reciprocal undo each other, leaving the base isolated.

Checking Solutions

Exponential equations demand verification, because the solving process can introduce values that fail in the original equation.

Domain validity is the first check. The base of an exponential function must be positive when the exponent is irrational, and nonzero when the exponent is negative. Any solution that violates these conditions is extraneous and must be discarded.

The substitution method introduces the most common source of false solutions. When

t = a^x

is used, only positive values of

t

are valid — because

a^x > 0

for any positive base

a

and any real

x

. A quadratic in

t

may produce a negative root, but that root has no corresponding real value of

x

.

Squaring both sides of an equation — sometimes necessary when rational exponents are involved — can also generate extraneous solutions. The equation

x^{1/2} = -3

has no real solution, since a square root is non-negative. But squaring gives

x = 9

, which does not satisfy the original equation.

Substituting each candidate back into the original equation is the reliable final step. If the left side equals the right side, the solution stands. If not, it is discarded — regardless of how cleanly the algebra produced it.