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 min18 problems
Practice Problems

Definition

An exponential equation is an equation where the unknown is in the exponent, for example 2x=322^x = 32 or 3x+1=813^{x+1} = 81.
The key idea is the same-base rule: if two powers with the same base are equal, their exponents are equal.
bm=bn  ⟹  m=n(b>0, b≠1)b^m = b^n \implies m = n \quad (b > 0,\ b \neq 1)
The same-base method:
  1. 1.Isolate the power: if a number multiplies the power, divide both sides by it (3⋅2x=48  ⟹  2x=163 \cdot 2^x = 48 \implies 2^x = 16).
  2. 2.Write both sides as powers of the same base (16=2416 = 2^4, 8=238 = 2^3, 4=224 = 2^2).
  3. 3.Set the exponents equal and solve the simpler equation.
  4. 4.Check by substituting the solution back into the original equation.
Useful rewrites:
  • Fractions become negative exponents: 127=3−3\frac{1}{27} = 3^{-3}
  • Roots become fractional exponents: 2=212\sqrt{2} = 2^{\frac{1}{2}}
  • Power of a power: 4x=(22)x=22x4^x = (2^2)^x = 2^{2x}
When the two sides cannot be written with the same base (like 2x=202^x = 20), you can estimate the solution with a table of powers: 24=162^4 = 16 and 25=322^5 = 32, so xx lies between 44 and 55. Logarithms, a later topic, give the exact value.

Try it now

Solve 2x=162^x = 16.

Worked Examples

Solve 2x=322^x = 32.
1

Write 32 as a power of 2

32=2⋅2⋅2⋅2⋅2=2532 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 2^5 → 2x=252^x = 2^5
2

Set the exponents equal

Same base 22, so the exponents must match → x=5x = 5
3

Check

25=322^5 = 32 ✓ → The solution is correct

Common Mistakes

Dividing by the base: 2x=32  ⟹  x=322=162^x = 32 \implies x = \frac{32}{2} = 16

Why it's wrong: 2x2^x means 22 multiplied by itself xx times, not 22 times xx. Dividing undoes multiplication, not exponentiation.

Correct: Write 3232 as a power of 22: 32=2532 = 2^5, so x=5x = 5. Check: 216=65536≠322^{16} = 65536 \neq 32.

Multiplying the factor into the base: 3⋅2x=48  ⟹  6x=483 \cdot 2^x = 48 \implies 6^x = 48

Why it's wrong: The exponent belongs only to the 22. By the order of operations, the power is calculated first and then multiplied by 33.

Correct: Divide first: 2x=16=242^x = 16 = 2^4, so x=4x = 4. Check: 3⋅24=3⋅16=483 \cdot 2^4 = 3 \cdot 16 = 48 ✓

Setting exponents equal when the bases differ: 4x=23  ⟹  x=34^x = 2^3 \implies x = 3

Why it's wrong: The same-base rule only works when the bases are really the same. 43=644^3 = 64, not 88.

Correct: First rewrite 4x=22x4^x = 2^{2x}, then 2x=32x = 3, so x=32x = \frac{3}{2}.

Losing the minus sign: 9x=127  ⟹  x=329^x = \frac{1}{27} \implies x = \frac{3}{2}

Why it's wrong: A number smaller than 11 needs a negative exponent when the base is bigger than 11: 127=3−3\frac{1}{27} = 3^{-3}.

Correct: 32x=3−33^{2x} = 3^{-3}, so x=−32x = -\frac{3}{2}. Quick test: 932=279^{\frac{3}{2}} = 27, which is far bigger than 127\frac{1}{27}.

Interactive Visual

Function graph

-10123456050100150xy(4.32, 100)

Tap or hover over the graph to read off values.

y = 5·2^xy = 100
  • Intersection point: (4.32, 100)
2³ = 8
3

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

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Practice Problems

18 problems
Problem 1 of 18
Easy
Solve 2x=162^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 88, 8181 and 127\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⋅2t=800100 \cdot 2^t = 800, so 2t=8=232^t = 8 = 2^3 and t=3t = 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⋅2t=4800300 \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⋅23=0.80.1 \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⋅2n=6.40.1 \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⋅25=320001000 \cdot 2^5 = 32000 views after 5 days and 1000⋅26=640001000 \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⋅2d=500001000 \cdot 2^d = 50000. First divide:

Key Takeaways

  • 1An exponential equation has the unknown in the exponent, like 2x=322^x = 32
  • 2Same-base rule: if bm=bnb^m = b^n (with b>0b > 0, b≠1b \neq 1), then m=nm = n
  • 3Isolate the power first: divide away any factor in front, as in 3⋅2x=48  ⟹  2x=163 \cdot 2^x = 48 \implies 2^x = 16
  • 4Rewrite with a common base using (bm)x=bmx(b^m)^x = b^{mx}, negative exponents for fractions (127=3−3\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 24=16<20<32=252^4 = 16 < 20 < 32 = 2^5, so the solution of 2x=202^x = 20 is between 44 and 55. Logarithms, which come later in the course, give the exact answer.
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 24=16<20<32=252^4 = 16 < 20 < 32 = 2^5, so the solution of 2x=202^x = 20 is between 44 and 55. Logarithms, which come later in the course, give the exact answer.
Because 1x=11^x = 1 for every xx. The equation 1x=151^x = 1^5 is true for all values of xx, so you cannot conclude that x=5x = 5. With any positive base other than 11, different exponents always give different results.
Yes. A power with a positive base is always positive, so 2x=−42^x = -4 and 2x=02^x = 0 have no solution. Before you start, check that the isolated power equals a positive number.
Yes, when the exponent contains x2x^2. For example 2x2=242^{x^2} = 2^4 gives x2=4x^2 = 4, so x=2x = 2 or x=−2x = -2. Check both in the original equation.

Glossary

Exponential equation
An equation in which the unknown appears in an exponent, such as 5x−1=1255^{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 22 for 4x=84^x = 8
Negative exponent
An exponent that means a reciprocal: b−n=1bnb^{-n} = \frac{1}{b^n}
Fractional exponent
An exponent that means a root: b1n=bnb^{\frac{1}{n}} = \sqrt[n]{b}, for example 2713=327^{\frac{1}{3}} = 3
Logarithm
The exponent you need to reach a number; it gives exact solutions such as the xx in 2x=202^x = 20 (a later topic)

Formula Card

Same-Base Rule

bm=bn  ⟹  m=nb^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

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

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

Negative Exponent

1bn=b−n\frac{1}{b^n} = b^{-n}

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

Fractional Exponent

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

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

Today's challengeOne problem, three tries

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