Exponential Equations with the Same Base

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Key Ideas Easy Medium Hard

If b^m = bⁿ, then m = n.

b is positive and b is not 1.

(b^k)ⁿ = b^kn

What to Remember

One base

Rewrite every exponential as a power of the same base.

4 = 2² and 8 = 2³. A fraction such as 18 is 2 to a negative power.

Multiply exponents

(b^k)^{x + n} = b to the power k(x + n).

Distribute that k.

Set the exponents equal

If b^m = bⁿ, and b is positive and not 1, then m = n.

Solve that linear equation.

Level 1

Easy

One side is already a power of the base you want

Example

Solve 5^{x − 1} = 5³.

Rewrite each side as a power of 5.

This side is already a power of 5.

5^{x − 1}

This side is already a power of 5.

5³

Same base: the exponents are equal.

x − 1 = 3

x = 4

Answer: x = 4

Solve 2^{x + 3} = 2⁴.

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Rewrite each side as a power of 2.

This side is already a power of 2.

2^{x + 3}

This side is already a power of 2.

2⁴

Same base: the exponents are equal.

x + 3 = 4

x = 1

Answer: x = 1

Solve 5^2x = 5³.

Show Answer

Rewrite each side as a power of 5.

This side is already a power of 5.

5^2x

This side is already a power of 5.

5³

Same base: the exponents are equal.

2x = 3

x = 32

Answer: x = 32

Solve 3^{x − 1} = 3⁴.

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Rewrite each side as a power of 3.

This side is already a power of 3.

3^{x − 1}

This side is already a power of 3.

3⁴

Same base: the exponents are equal.

x − 1 = 4

x = 5

Answer: x = 5

Solve 4^x = 4³.

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Rewrite each side as a power of 4.

This side is already a power of 4.

4^x

This side is already a power of 4.

4³

Same base: the exponents are equal.

x = 3

Answer: x = 3

Solve 9^x = 3⁴.

Show Answer

Rewrite each side as a power of 3.

9 = 3²

9^x = (3²)^x

Multiply the exponents.

2(x) = 2x

3^2x

This side is already a power of 3.

3⁴

Same base: the exponents are equal.

2x = 4

x = 2

Answer: x = 2

Level 2

Medium

Both sides need a new base

Example

Solve 4^{x + 1} = 8^{x − 2}.

Rewrite each side as a power of 2.

4 = 2²

4^{x + 1} = (2²)^{x + 1}

Multiply the exponents.

2(x + 1) = 2x + 2

2^{2x + 2}

8 = 2³

8^{x − 2} = (2³)^{x − 2}

Multiply the exponents.

3(x − 2) = 3x − 6

2^{3x − 6}

Same base: the exponents are equal.

2x + 2 = 3x − 6

−x = −8

x = 8

Answer: x = 8

Solve 9^{x + 1} = 27^{x − 1}.

Show Answer

Rewrite each side as a power of 3.

9 = 3²

9^{x + 1} = (3²)^{x + 1}

Multiply the exponents.

2(x + 1) = 2x + 2

3^{2x + 2}

27 = 3³

27^{x − 1} = (3³)^{x − 1}

Multiply the exponents.

3(x − 1) = 3x − 3

3^{3x − 3}

Same base: the exponents are equal.

2x + 2 = 3x − 3

−x = −5

x = 5

Answer: x = 5

Solve 8^{2x + 1} = 4^{x + 5}.

Show Answer

Rewrite each side as a power of 2.

8 = 2³

8^{2x + 1} = (2³)^{2x + 1}

Multiply the exponents.

3(2x + 1) = 6x + 3

2^{6x + 3}

4 = 2²

4^{x + 5} = (2²)^{x + 5}

Multiply the exponents.

2(x + 5) = 2x + 10

2^{2x + 10}

Same base: the exponents are equal.

6x + 3 = 2x + 10

4x = 7

x = 74

Answer: x = 74

Solve 25^{x − 2} = 5^3x.

Show Answer

Rewrite each side as a power of 5.

25 = 5²

25^{x − 2} = (5²)^{x − 2}

Multiply the exponents.

2(x − 2) = 2x − 4

5^{2x − 4}

This side is already a power of 5.

5^3x

Same base: the exponents are equal.

2x − 4 = 3x

−x = 4

x = −4

Answer: x = −4

Solve 16^{x + 1} = 8^2x.

Show Answer

Rewrite each side as a power of 2.

16 = 2⁴

16^{x + 1} = (2⁴)^{x + 1}

Multiply the exponents.

4(x + 1) = 4x + 4

2^{4x + 4}

8 = 2³

8^2x = (2³)^2x

Multiply the exponents.

3(2x) = 6x

2^6x

Same base: the exponents are equal.

4x + 4 = 6x

−2x = −4

x = 2

Answer: x = 2

Solve 27^{x + 2} = 9^{2x + 1}.

Show Answer

Rewrite each side as a power of 3.

27 = 3³

27^{x + 2} = (3³)^{x + 2}

Multiply the exponents.

3(x + 2) = 3x + 6

3^{3x + 6}

9 = 3²

9^{2x + 1} = (3²)^{2x + 1}

Multiply the exponents.

2(2x + 1) = 4x + 2

3^{4x + 2}

Same base: the exponents are equal.

3x + 6 = 4x + 2

−x = −4

x = 4

Answer: x = 4

Level 3

Hard

A fraction base, or a product of exponentials

Example

Solve 16^{x − 1} = (12)^3x.

Rewrite each side as a power of 2.

16 = 2⁴

16^{x − 1} = (2⁴)^{x − 1}

Multiply the exponents.

4(x − 1) = 4x − 4

2^{4x − 4}

12 = 2⁻¹

(12)^3x = (2⁻¹)^3x

Multiply the exponents.

−1(3x) = −3x

2^−3x

Same base: the exponents are equal.

4x − 4 = −3x

7x = 4

x = 47

Answer: x = 47

Solve 4^{x + 2} = 18.

Show Answer

Rewrite each side as a power of 2.

4 = 2²

4^{x + 2} = (2²)^{x + 2}

Multiply the exponents.

2(x + 2) = 2x + 4

2^{2x + 4}

18 = 2⁻³

Same base: the exponents are equal.

2x + 4 = −3

2x = −7

x = −72

Answer: x = −72

Solve 9^{2x − 1} = 27^{x + 2}.

Show Answer

Rewrite each side as a power of 3.

9 = 3²

9^{2x − 1} = (3²)^{2x − 1}

Multiply the exponents.

2(2x − 1) = 4x − 2

3^{4x − 2}

27 = 3³

27^{x + 2} = (3³)^{x + 2}

Multiply the exponents.

3(x + 2) = 3x + 6

3^{3x + 6}

Same base: the exponents are equal.

4x − 2 = 3x + 6

x = 8

Answer: x = 8

Solve (13)^{x − 4} = 9^{x + 1}.

Show Answer

Rewrite each side as a power of 3.

13 = 3⁻¹

(13)^{x − 4} = (3⁻¹)^{x − 4}

Multiply the exponents.

−1(x − 4) = −x + 4

3^{−x + 4}

9 = 3²

9^{x + 1} = (3²)^{x + 1}

Multiply the exponents.

2(x + 1) = 2x + 2

3^{2x + 2}

Same base: the exponents are equal.

−x + 4 = 2x + 2

−3x = −2

x = 23

Answer: x = 23

Solve 8^{x − 1} · 4^{x + 2} = 2⁷.

Show Answer

Rewrite 8 and 4 as powers of 2.

8 = 2³ and 4 = 2²

8^{x − 1} = 2^{3x − 3}

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

Product rule: add the exponents.

2^{3x − 3} · 2^{2x + 4} = 2^{5x + 1}

2^{5x + 1} = 2⁷

Same base: the exponents are equal.

5x + 1 = 7

5x = 6

x = 65

Answer: x = 65

Solve 5^{2x + 1} = 1125.

Show Answer

Rewrite each side as a power of 5.

This side is already a power of 5.

5^{2x + 1}

1125 = 5⁻³

Same base: the exponents are equal.

2x + 1 = −3

2x = −4

x = −2

Answer: x = −2

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