Solving Exponential Equations Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Solving Exponential Equations.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Solving exponential equations means finding the unknown variable trapped in an exponent by applying logarithms to both sides, using the power rule to bring the exponent down, and then isolating the variable with standard algebra.

When the variable is trapped in an exponent, logarithms free it. Taking $\log$ of both sides brings the exponent down to ground level where you can solve for it using algebra.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Logarithms pull the unknown exponent down to ground level where algebra can isolate it.

Common stuck point: The procedure for solving exponential equations is the easy part; the trap is dividing instead of taking a log. Asking "Is the unknown stuck up in the exponent, so I need a logarithm to bring it down?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the unknown stuck up in the exponent, so I need a logarithm to bring it down?

Worked Examples

Example 1

easy

Solve

3^x = 81

Answer

x = 4

First step

Recognize that

81

is a power of

3

81 = 3^4

Full solution

2
So the equation becomes $3^x = 3^4$ .
3
Since the bases are equal, the exponents must be equal: $x = 4$ .

When both sides of an exponential equation can be written with the same base, set the exponents equal. This is the simplest method and avoids logarithms entirely. Always check if the number is a recognizable power of the base.

Example 2

medium

Solve

5^{2x-1} = 12

Example 3

easy

Solve

2^{x-1} = 16

by matching bases.

Example 4

medium

A bacteria culture follows

N = 200\cdot 3^t

(hours). After how many hours does

N = 5400

Example 5

hard

Solve

3^{2x+1} = 5^{x}

, exact form.

Practice Problems

3^{2x} = 27

Example 11

medium

Solve

2^x = 5

P = 100\cdot 2^t

. When does

P = 1600

Example 18

challenge

Solve

4^x - 5\cdot 2^x + 4 = 0

Example 19

challenge

Solve

2^x = 3^{x-2}

for

x

in log form.

Example 20

challenge

Solve

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

Example 21

medium

Solve

6^x = 36^{x-1}

Example 22

medium

Solve

e^x = 10

3^x = 20

, exact form.

Example 29

medium

Solve

7\cdot 2^x = 56

Example 30

medium

Solve

2^{3x} = 5

Example 31

medium

Solve

5^{x+2} = 3^{x-1}

, exact form.

Example 32

medium

Solve

4^{x+1} = 2^{x+5}

Example 33

medium

A loan grows by

A(t) = 1000\cdot 1.05^t

. After how many full years does it first exceed

\$1500

? Give exact

t

Example 34

medium

Solve

e^{x+1} = 4

Example 35

medium

Solve

2\cdot 5^{x} = 50

Example 36

hard

Solve

9^x - 6\cdot 3^x - 27 = 0

Example 37

hard

Solve

e^{2x} + e^x - 12 = 0

Example 38

hard

Carbon-14 has half-life

5730

years. A sample retains

30\%

of its

^{14}\text{C}

. Find the age

t

Example 39

hard

Solve

2^{x^2 - 1} = 8

Example 40

hard

Solve

2^{2x} + 2^{x+2} - 32 = 0

Example 41

challenge

Solve

4^x + 6^x = 9^x

Example 42

challenge

Find all real

x

with

2^x \cdot 5^x = 10^4

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Related Concepts

Exponential Function Logarithm Logarithm Properties Solving Logarithmic Equations

Background Knowledge

These ideas may be useful before you work through the harder examples.

exponential functionlogarithmlogarithm properties