Why must the base not be 1?

Because $1^x = 1$ for every $x$. The equation $1^x = 1^5$ is true for all values of $x$, so you cannot conclude that $x = 5$. With any positive base other than $1$, different exponents always give different results.

Can an exponential equation have no solution?

Yes. A power with a positive base is always positive, so $2^x = -4$ and $2^x = 0$ have no solution. Before you start, check that the isolated power equals a positive number.

Can there be two solutions?

Yes, when the exponent contains $x^2$. For example $2^{x^2} = 2^4$ gives $x^2 = 4$, so $x = 2$ or $x = -2$. Check both in the original equation.

Teacher Resources

Solving Exponential Equations

Solve equations with the variable in the exponent by writing both sides as powers of the same base, and estimate solutions that are not whole numbers.

Advanced25 minLesson

Printable worksheet

Definition

An exponential equation is an equation where the unknown is in the exponent, for example

2^x = 32

3^{x+1} = 81

The key idea is the same-base rule: if two powers with the same base are equal, their exponents are equal.

b^m = b^n \implies m = n \quad (b > 0,\ b \neq 1)

The same-base method:

1.Isolate the power: if a number multiplies the power, divide both sides by it ( $3 \cdot 2^x = 48 \implies 2^x = 16$ ).
2.Write both sides as powers of the same base ( $16 = 2^4$ , $8 = 2^3$ , $4 = 2^2$ ).
3.Set the exponents equal and solve the simpler equation.
4.Check by substituting the solution back into the original equation.

Useful rewrites:

Fractions become negative exponents: $\frac{1}{27} = 3^{-3}$
Roots become fractional exponents: $\sqrt{2} = 2^{\frac{1}{2}}$
Power of a power: $4^x = (2^2)^x = 2^{2x}$

When the two sides cannot be written with the same base (like

2^x = 20

), you can estimate the solution with a table of powers:

2^4 = 16

and

2^5 = 32

, so

x

lies between

4

and

5

. Logarithms, a later topic, give the exact value.

Try it now

Solve

2^x = 16

Worked Examples

Solve $2^x = 32$ .

Write 32 as a power of 2

$32 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 2^5$ → $2^x = 2^5$

Set the exponents equal

Same base $2$ , so the exponents must match → $x = 5$

Check

$2^5 = 32$ ✓ → The solution is correct

Answer: $x = 5$

Common Mistakes

Dividing by the base: $2^x = 32 \implies x = \frac{32}{2} = 16$

Why it's wrong: $2^x$ means $2$ multiplied by itself $x$ times, not $2$ times $x$ . Dividing undoes multiplication, not exponentiation.

Correct: Write $32$ as a power of $2$ : $32 = 2^5$ , so $x = 5$ . Check: $2^{16} = 65536 \neq 32$ .

Multiplying the factor into the base: $3 \cdot 2^x = 48 \implies 6^x = 48$

Why it's wrong: The exponent belongs only to the $2$ . By the order of operations, the power is calculated first and then multiplied by $3$ .

Correct: Divide first: $2^x = 16 = 2^4$ , so $x = 4$ . Check: $3 \cdot 2^4 = 3 \cdot 16 = 48$ ✓

Setting exponents equal when the bases differ: $4^x = 2^3 \implies x = 3$

Why it's wrong: The same-base rule only works when the bases are really the same. $4^3 = 64$ , not $8$ .

Correct: First rewrite $4^x = 2^{2x}$ , then $2x = 3$ , so $x = \frac{3}{2}$ .

Losing the minus sign: $9^x = \frac{1}{27} \implies x = \frac{3}{2}$

Why it's wrong: A number smaller than $1$ needs a negative exponent when the base is bigger than $1$ : $\frac{1}{27} = 3^{-3}$ .

Correct: $3^{2x} = 3^{-3}$ , so $x = -\frac{3}{2}$ . Quick test: $9^{\frac{3}{2}} = 27$ , which is far bigger than $\frac{1}{27}$ .

Interactive Visual

Function graph

Tap or hover over the graph to read off values.

y = 5·2^xy = 100

Intersection point: (4.32, 100)

Base:

Calculate:2³ = 8

See how powers of a number grow on the number line. Change the base and exponent.

Interactive Sandbox

Expression Calculator

Try these:

History

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Practice Problems

18 problems

Problem 1 of 18

Easy

Solve $2^x = 16$ .

Why It Matters

Exponential equations answer the question "when?" for anything that grows or shrinks by a constant factor:

Biology: When will a bacteria colony that doubles every hour reach 4800 cells?
Finance: How many years until an investment doubles or triples?
Technology: How many doublings of storage are needed to hold a given amount of data?
Science: How long until a substance drops to a safe level?

In the previous lessons you calculated the amount for a given time. Here you turn the question around and find the time (the exponent) for a given amount. The same-base method also trains you to see numbers like

8

81

and

\frac{1}{27}

as powers, a skill you will use again with logarithms.

Real World Applications

Bacteria Colonies

A colony that doubles every hour follows $N = N_0 \cdot 2^t$. Solving for $t$ tells a lab when the colony reaches a target size.

Example:

A colony of 100 cells doubles every hour. It reaches 800 cells when $100 \cdot 2^t = 800$ , so $2^t = 8 = 2^3$ and $t = 3$ hours.

1Try It Yourself

A sample starts with 300 bacteria and doubles every hour.

After how many hours are there 4800 bacteria?

Step 1: Write the mathematical expression

Solve $300 \cdot 2^t = 4800$ . First divide:

Folding Paper

Each fold doubles the thickness of a sheet of paper, so the thickness after $n$ folds is $T = T_0 \cdot 2^n$.

Example:

A sheet 0.1 mm thick is 0.8 mm thick after 3 folds, because $0.1 \cdot 2^3 = 0.8$ .

2Try It Yourself

A sheet of paper is 0.1 mm thick. Every fold doubles the thickness.

How many folds make the stack 6.4 mm thick?

Step 1: Write the mathematical expression

Solve $0.1 \cdot 2^n = 6.4$ . First divide:

Viral Videos

Views that double every day grow like $V = V_0 \cdot 2^d$. When the target is not a power of 2, a table of powers shows the day it is passed.

Example:

A video with 1000 views that doubles daily has $1000 \cdot 2^5 = 32000$ views after 5 days and $1000 \cdot 2^6 = 64000$ after 6 days.

3Try It Yourself

A video has 1000 views and its views double every day.

Between which two whole days does it pass 50000 views?

Step 1: Write the mathematical expression

Solve $1000 \cdot 2^d = 50000$ . First divide:

Key Takeaways

1An exponential equation has the unknown in the exponent, like $2^x = 32$
2Same-base rule: if $b^m = b^n$ (with $b > 0$ , $b \neq 1$ ), then $m = n$
3Isolate the power first: divide away any factor in front, as in $3 \cdot 2^x = 48 \implies 2^x = 16$
4Rewrite with a common base using $(b^m)^x = b^{mx}$ , negative exponents for fractions ( $\frac{1}{27} = 3^{-3}$ ) and fractional exponents for roots
5Always check the solution by substituting it back into the original equation
6If no common base exists, a table of powers shows between which integers the solution lies; logarithms give the exact value

Frequently Asked Questions

Then the same-base method does not apply. You can still estimate: make a table of powers and find where the right side fits, for example

2^4 = 16 < 20 < 32 = 2^5

, so the solution of

2^x = 20

is between

4

and

5

. Logarithms, which come later in the course, give the exact answer.

Glossary

Exponential equation: An equation in which the unknown appears in an exponent, such as $5^{x-1} = 125$
Same-base method: Solving an exponential equation by writing both sides as powers of one base and setting the exponents equal
Common base: A base that both sides can be written with, like $2$ for $4^x = 8$
Negative exponent: An exponent that means a reciprocal: $b^{-n} = \frac{1}{b^n}$
Fractional exponent: An exponent that means a root: $b^{\frac{1}{n}} = \sqrt[n]{b}$ , for example $27^{\frac{1}{3}} = 3$
Logarithm: The exponent you need to reach a number; it gives exact solutions such as the $x$ in $2^x = 20$ (a later topic)

Formula Card

Same-Base Rule

b^m = b^n \implies m = n

Valid for $b > 0$ and $b \neq 1$. Write both sides with the same base, then set the exponents equal.

Power of a Power

(b^m)^x = b^{mx}

Used to change the base, for example $8^x = (2^3)^x = 2^{3x}$.

Negative Exponent

\frac{1}{b^n} = b^{-n}

Turns a fraction into a power, for example $\frac{1}{16} = 2^{-4}$.

Fractional Exponent

\sqrt[n]{b^m} = b^{\frac{m}{n}}

Turns a root into a power, for example $\sqrt{2} = 2^{\frac{1}{2}}$.

Solving Exponential Equations

Definition

Worked Examples

Common Mistakes

Interactive Visual

Function graph

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Bacteria Colonies

Folding Paper

Viral Videos

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Same-Base Rule

Power of a Power

Negative Exponent

Fractional Exponent

More in This Topic

Introduction to Exponential Functions

Exponential Growth

Graphing Exponential Functions

Exponential Decay