Exponential equations: a common base and complete solution sets

2026-10-22 huynhvietkhanh Comments: 0

Reduce equations to a common base, use reciprocal conjugates, and retain every zero-product branch.

High schoolPowers & Logarithms

Content updated: October 22, 2026

Two powers with the same positive base different from 1 are equal exactly when their exponents are equal. First reduce correctly to a common base; for a product equation, retain every branch that makes the product 0.

Objectives: identify a common base, use reciprocal conjugates and factorization, and check conditions before division.

Prerequisites: real powers, logarithms, and algebraic equations.

On this page: Idea · Method · Examples · Practice · Further learning

1. Injectivity of exponential functions

For $a>0$, $a\ne1$, and expressions $u(x),v(x)$ taking real values on the domain under consideration, we have

$$a^{u(x)}=a^{v(x)}\quad\Longleftrightarrow\quad u(x)=v(x).$$

This holds because $a^t$ is strictly monotone: increasing for $a>1$ and decreasing for $0<a<1$. For equality, both base cases imply the same conclusion about the exponents. If $a=1$, both sides always equal 1, so $u=v$ does not follow.

Powers with positive bases are always different from 0. Thus we may divide by $a^{u(x)}$ wherever the exponent is defined. If a base or exponent contains a variable in a denominator or root, determine the domain first.

2. Reducing to a common base

Use $(ab)^x=a^xb^x$, $a^{x+c}=a^c a^x$, or $(a/b)^x=a^x/b^x$ with positive bases. For reciprocal bases, express one using the negative of the exponent. For an equation $a^u=c$, require $c>0$ and then write $c=a^{\log_a c}$.

A sum of two powers cannot always be combined into one power: $a^x+b^x$ generally differs from $(a+b)^x$. For an equation involving polynomials and exponentials, collect terms to find a common factor. If $P(x)Q(x)=0$, take the union of the solutions of $P=0$ or $Q=0$; do not discard a factor by division without checking it.

3. Five groups of worked examples

Example 1. Three common-base equations

(a) $3^x2^x-6=0$ is equivalent to $6^x=6^1$, so $x=1$. (b) $2^x-3=0$ implies $x=\log_2 3$, since 3 is positive. (c) $2^x3^x=36=6^2$ implies $x=2$. All expressions are defined on $\mathbb R$.

Example 2. Grouping exponents differing by a constant

Solve $2^{x+1}+2^x-3=0$. Write $2^{x+1}=2\cdot2^x$ to obtain $3\cdot2^x=3$, so $2^x=1=2^0$ and $x=0$. Substitution gives $2+1-3=0$.

Example 3. Different bases with a common exponent

Solve $3^{x+1}-2^{x+1}=0$. Divide by $2^{x+1}>0$ to obtain $(3/2)^{x+1}=1=(3/2)^0$. Since base 3/2 differs from 1, $x+1=0$, or $x=-1$.

Example 4. Reciprocal conjugate bases

Solve $(\sqrt2-1)^x=(\sqrt2+1)^x$. Both bases are positive and $(\sqrt2-1)(\sqrt2+1)=1$. Set $a=\sqrt2-1\in(0;1)$; the equation becomes $a^x=a^{-x}$. Injectivity implies $x=-x$, so $x=0$. Multiplying the conjugates establishes the reciprocal relation between the bases.

Example 5. Combining a polynomial with an exponential

Solve $4x^2+x3^x+3^{x+1}=2x^2 3^x+2x+6$. Bring all terms to one side and group those involving $3^x$:

$$\begin{aligned}(2x^2-x-3)3^x-2(2x^2-x-3)&=0,\\(2x^2-x-3)(3^x-2)&=0.\end{aligned}$$

The polynomial factors as $(2x-3)(x+1)$, so the first branch has roots $3/2$ and $-1$. The other branch, $3^x=2$, gives $x=\log_3 2$. Hence the solution set is $\{-1;3/2;\log_3 2\}$. Each branch makes a factor 0, so all three satisfy the original equation.

4. Practice exercises

Solve $4^{x-1}=2^{x+2}$.
Solve $(1/3)^{2x-1}=3$.
Solve $(x-2)(2^x-4)=0$.

Show hints and answers

Using $4=2^2$ gives $2x-2=x+2$, so $x=4$.
Write $(1/3)^{2x-1}=3^{1-2x}$; then $1-2x=1$ and $x=0$.
Both branches give $x=2$, so the solution set has just one element, $\{2\}$; do not count a repeated solution as two values.

5. Common mistakes

Do not equate exponents before obtaining the same base. Do not use injectivity for base 1. Do not combine a sum by adding the bases. Do not lose polynomial-factor roots by division; combine repeated solutions at the end.