Exponential Equations with the Same Base

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Key Ideas Easy Medium Hard Super Challenge

What to Remember

Same base

If a^m = aⁿ and a > 0, a ≠ 1, then m = n.

Rewrite first

Write 4 as 2², 8 as 2³, 9 as 3², 27 as 3³.

Then equate exponents

After both sides are the same base, the exponents are equal. Solve that linear equation.

Check

Substitute back into the original exponential equation.

Level 1

Easy

Both sides already the same base.

Example

Solve. 2^x = 2⁵

The bases match, so the exponents match.

x = 5

Check: 2⁵ = 32.

Answer: x = 5

Solve. 3^x+1 = 3⁴

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x + 1 = 4

x = 3

Answer: x = 3

Solve. 5^2x = 5⁶

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2x = 6

x = 3

Answer: x = 3

Solve. 7^x = 7

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7 = 7¹

x = 1

Answer: x = 1

Solve. 2^x = 2^x

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True for every x. Infinitely many solutions.

Answer: all real x

Solve. 4^x = 4⁰

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4⁰ = 1

x = 0

Answer: x = 0

Level 2

Medium

Rewrite one side as a power of the same base.

Example

Solve. 2^x = 8

8 = 2³

2^x = 2³

x = 3

Answer: x = 3

Solve. 3^x = 81

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81 = 3⁴

x = 4

Answer: x = 4

Solve. 5^x+2 = 25

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25 = 5²

x + 2 = 2

x = 0

Answer: x = 0

Solve. 9^x = 3⁶

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9 = 3²

(3²)^x = 3⁶

3^2x = 3⁶

2x = 6

x = 3

Answer: x = 3

Solve. 2^x = 18

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18 = 2⁻³

x = −3

Answer: x = −3

Solve. 4^x = 2

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4 = 2²

2^2x = 2¹

2x = 1

x = 12

Answer: 12

Level 3

Hard

Both sides need a rewrite.

Example

Solve. 4^x+1 = 8^x−1

4 = 2², 8 = 2³

(2²)^x+1 = (2³)^x−1

2^2x+2 = 2^3x−3

2x + 2 = 3x − 3

x = 5

Check: 4⁶ = 4096 and 8⁴ = 4096.

Answer: x = 5

Solve. 9^x = 27^x−1

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9 = 3², 27 = 3³

3^2x = 3^3x−3

2x = 3x − 3

x = 3

Answer: x = 3

Solve. 16 × 2^x = 2⁷

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16 = 2⁴

2^4+x = 2⁷

x = 3

Answer: x = 3

Solve. 8^x = 4^x+3

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2^3x = 2^2x+6

3x = 2x + 6

x = 6

Answer: x = 6

Solve. 3^2x−1 = 127

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127 = 3⁻³

2x − 1 = −3

2x = −2

x = −1

Answer: x = −1

Solve. 25^x = 125^x−1. 25 and 125 are both powers of 5.

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25 = 5², 125 = 5³

5^2x = 5^3x−3

2x = 3x − 3

x = 3

Answer: x = 3

Super Challenge

Solve. 81 × 13^t = 3.

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81 = 3⁴, 13 = 3⁻¹

3⁴ × 3^-t = 3¹

3^{4 − t} = 3¹

t = 3

Answer: t = 3

Find every x so that 4^x = 2^2x. Then find x so that 4^x = 2^x.

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First: 2^2x = 2^2x, always true.

Second: 2^2x = 2^x, so 2x = x, x = 0.

Answer: all x; x = 0

A student solves 2^x = 8^x by writing x = x and saying all real x. Correct the work.

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8 = 2³, so 2^x = 2^3x

x = 3x

0 = 2x

x = 0

Check: 1 = 1.

Answer: x = 0

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