Grade by Grade
A. Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Solving Exponential Equations with a Common Base

A3.2 A3.1About 70 minutes
Learning goals
  • I can explain why bᵐ = bⁿ means m = n (for b > 0, b ≠ 1).
  • I can solve exponential equations by writing both sides as powers of the same base.
  • I can solve equations that lead to linear or quadratic equations in the exponents.
  • I can factor out a common power, as in 3^(x + 2) − 3ˣ = 72, before solving.

Words to know

common base
A base that both sides of an equation can be written in, such as 2 for 8ˣ = 32.
one-to-one property
For b > 0, b ≠ 1: if bᵐ = bⁿ, then m = n. Different exponents always give different powers.
exponential equation
An equation with the variable in an exponent.
extraneous solution
A value that comes out of the algebra but does not satisfy the original equation.

When the bases can match

Some exponential equations hide a neat answer. In 9ˣ = 27^(x − 1), both 9 and 27 are powers of 3. Rewrite them and the equation becomes a simple linear equation.

Remember from Grade 11: in Exponential Functions (MCR3U) you rewrote powers with different bases, such as 9ˣ = 3^(2x), and solved simple equations like 4ˣ = 32 by writing both sides with a common base. Now you will solve harder ones systematically, and in lesson 12 you will handle the ones where no common base exists.

Why matching exponents works

The function y = bˣ (b > 0, b ≠ 1) is always increasing or always decreasing, so it never takes the same value twice. That makes it one-to-one:

If bᵐ = bⁿ, then m = n.

In logarithm language: take log_b of both sides, and log_b (bᵐ) = m, log_b (bⁿ) = n.

So the method is:

  1. Write each side as a single power of the same base.
  2. Set the exponents equal.
  3. Solve the resulting equation and check.
power 1power 2power 3power 4power 5power 6power 7power 8base 2248163264128256base 3392781243729base 5525125625base 1010100100010 000
Powers of 2, 3, 5 and 10. Knowing these families on sight is the key to spotting a common base: 8, 32 and 128 are all powers of 2; 27 and 243 are powers of 3.
Worked example

Example 1: One side is already a power

Solve 2^(x + 3) = 32.

  1. 32 = 2⁵, so 2^(x + 3) = 2⁵.
  2. Equate exponents: x + 3 = 5.
  3. x = 2.
  4. Check: 2^(2 + 3) = 2⁵ = 32 ✓

x = 2

Worked example

Example 2: Both sides need rewriting

Solve 9ˣ = 27^(x − 1).

  1. 9 = 3² and 27 = 3³.
  2. (3²)ˣ = (3³)^(x − 1), so 3^(2x) = 3^(3x − 3).
  3. 2x = 3x − 3, so x = 3.
  4. Check: 9³ = 729 and 27² = 729 ✓

x = 3

Worked example

Example 3: Fractions and negative exponents

Solve (1/4)ˣ = 8.

  1. 1/4 = 2⁻² and 8 = 2³.
  2. (2⁻²)ˣ = 2³, so 2^(−2x) = 2³.
  3. −2x = 3, so x = −3/2.
  4. Check: (1/4)^(−3/2) = 4^(3/2) = (√4)³ = 8 ✓

x = −3/2

xy−3−2−11224681012141618(−1.5, 8)(−1, 4)(−2, 16)y = 8y = (1/4)ˣ
The decreasing curve y = (1/4)ˣ meets the line y = 8 at x = −1.5, confirming the algebra in Example 3.
Worked example

Example 4: A quadratic in the exponent

Solve 5^(x² − x) = 25.

  1. 25 = 5², so x² − x = 2.
  2. x² − x − 2 = 0, so (x − 2)(x + 1) = 0.
  3. x = 2 or x = −1.
  4. Check x = 2: 5^(4 − 2) = 5² = 25 ✓. Check x = −1: 5^(1 + 1) = 25 ✓

x = 2 or x = −1

Worked example

Example 5: Products of powers

Solve 16ˣ × 4^(x − 1) = 64.

  1. Write everything in base 2: 2^(4x) × 2^(2x − 2) = 2⁶.
  2. Product of powers: 2^(6x − 2) = 2⁶.
  3. 6x − 2 = 6, so x = 4/3.
  4. Check: 16^(4/3) × 4^(1/3) = 2^(16/3) × 2^(2/3) = 2^(18/3) = 2⁶ = 64 ✓

x = 4/3

Worked example

Example 6: Factor out a common power

Solve 3^(x + 2) − 3ˣ = 72.

  1. You cannot combine 3^(x + 2) − 3ˣ into one power by subtracting exponents. Instead, note 3^(x + 2) = 3ˣ × 3² = 9(3ˣ).
  2. So 9(3ˣ) − 3ˣ = 72, which is 8(3ˣ) = 72.
  3. 3ˣ = 9 = 3², so x = 2.
  4. Check: 3⁴ − 3² = 81 − 9 = 72 ✓

x = 2

Know your powers

Keep the families in your head: powers of 2 up to 2¹⁰ = 1024, powers of 3 up to 3⁶ = 729, powers of 5 up to 5⁴ = 625. Also remember √2 = 2^(1/2), ∛4 = 2^(2/3) and 1/8 = 2⁻³.

Just for fun

2¹⁰ = 1024 is so close to 1000 that, for decades, computer people called 1024 bytes a "kilobyte". Today the official name for 1024 bytes is a kibibyte, which sounds like a snack for a small dog.

Hands-on activity

Equation swap

You need: paper, a pencil and a partner (or do both parts yourself, a day apart).

  1. Copy the power families from the diagram onto a scrap of paper.
  2. Write three exponential equations for your partner to solve. Build them backwards: choose an answer first (for example x = 3), choose a base (2), and then disguise both sides, e.g. 4^(x − 1) = 2^(x + 1) (check: 4² = 16 and 2⁴ = 16 ✓).
  3. Make one equation linear in the exponent, one quadratic (like 2^(x²) = 16), and one with a fraction base (like (1/9)ˣ = 27).
  4. Swap, solve, and check each other's answers by substitution.

What to look for: every equation must have both sides written in the same base before the exponents are set equal. For 2^(x²) = 16, the answer is x = ±2 (both). For (1/9)ˣ = 27, 3^(−2x) = 3³ gives x = −3/2.

Your turn

1

Solve 3^(2x − 1) = 243.

2

Solve 4ˣ = 8.

3

Solve 25^(x + 1) = 125ˣ.

4

Solve (1/2)ˣ = 16.

5

Solve 10^(x²) = 10 000.

6

Solve 2^(x + 1) + 2ˣ = 48.

7

Solve 8^(x − 2) = 4^(x + 1).

More practice: warm-up

1

Solve 2ˣ = 64.

2

Solve 3ˣ = 1/9.

3

Solve 5^(x − 1) = 25.

4

Solve 10ˣ = 0.001.

5

Solve 7^(2x) = 1.

More practice: on your own

1

Solve 27^(x + 1) = 9^(2x).

2

Solve 4^(x²) = 2^(x + 3).

3

Solve √(2ˣ) = 8.

4

Solve 5^(x + 2) − 5ˣ = 600.

5

Solve (1/8)^(x − 1) = 32^(x + 3).

Challenge

1

Solve 2^(2x) − 12(2ˣ) + 32 = 0. (Hint: let u = 2ˣ.)

2

Solve 9ˣ − 3^(x + 1) − 54 = 0.

3

Solve 3^(x²) × 9ˣ = 27³ (hint: 27³ = 3⁹).

Check your understanding

Quiz

1.

Solve 2^(x − 1) = 16.

2.

Which equation is 4^(x + 1) = 8ˣ rewritten with base 2?

3.

Solve 9ˣ = 1/3.

4.

What are the solutions of 2^(x²) = 2^(3x + 4)?

5.

What is the first step to solve 2^(x + 3) − 2ˣ = 56?

6.

Why can you set the exponents equal when bᵐ = bⁿ?

For parents and teachers

Big idea (A3.2): if both sides of an exponential equation can be written with the same base, the exponents must be equal. That turns the problem into a linear or quadratic equation.

How to help: Before your teen starts, ask "What base can both sides be written in?" Quiz the power families in the car or at dinner ("What power of 2 is 64?" "What power of 3 is 81?").

Common difficulties:

  • Forgetting brackets when raising a power to a power: (3²)^(x − 1) = 3^(2x − 2), not 3^(2x − 1).
  • Subtracting exponents across a minus sign: 3^(x + 2) − 3ˣ is not 3².
  • Missing the second solution of a quadratic in the exponent, or missing a negative one.
  • Keeping solutions like 3ˣ = −6, which has no real solution because powers are positive.