Grade by Grade
B. Exponential Functions · Exponential Functions

Same Function, Different Bases

B2.4About 65 minutes
Learning goals
  • I can rewrite an exponential function using a different base, such as 9ˣ = 3^(2x).
  • I can explain why two forms are equivalent using exponent laws, transformations and graphs.
  • I can show that a horizontal translation of an exponential graph is also a vertical stretch.
  • I can solve simple exponential equations by writing both sides with a common base.

Words to know

equivalent functions
Functions that give the same output for every input, so their graphs are identical, even if their equations look different.
common base
A single base that several numbers can be written as powers of. 4, 8, 16 and 32 all have the common base 2.
exponential equation
An equation with the variable in an exponent, such as 4ˣ = 32.

One function, many names

Is y = 9ˣ the same function as y = 3^(2x)? Make a table: at x = 1 both give 9; at x = 0.5 both give 3; at x = −1 both give 1/9. They agree everywhere, so their graphs are identical. The same function can be written with different bases. Spotting this lets you compare functions, rewrite models in a more useful form, and even solve some exponential equations without guessing.

Three ways to see that 9ˣ = 3^(2x)

  1. Exponent laws: 9 = 3², so 9ˣ = (3²)ˣ = 3^(2x) by the power of a power law.
  2. Transformations: y = 3^(2x) is y = 3ˣ compressed horizontally by a factor of 1/2. Every point gets twice as close to the y-axis: (1, 3) → (0.5, 3) and (2, 9) → (1, 9). Those are exactly the points of y = 9ˣ.
  3. Graphs and tables (technology): graph y = 9ˣ and y = 3^(2x) in Desmos. Only one curve appears, because they lie exactly on top of each other. The tables match row by row.
xy−2−10120123456789× 1/2(1, 3)(0.5, 3)y = 9ˣ = 3^(2x)y = 3ˣ
Compressing y = 3ˣ (dashed) horizontally by 1/2 moves (2, 9) to (1, 9) and (1, 3) to (0.5, 3). The result is exactly y = 9ˣ: the same function as y = 3^(2x).

Rewriting with a common base

To rewrite, express each base as a power of the same number. Knowing the powers of 2 and 3 makes this fast.

2= 2¹4= 2²8= 2³16= 2⁴32= 2⁵64= 2⁶3= 3¹9= 3²27= 3³81= 3⁴Also: 1/2 = 2⁻¹,1/9 = 3⁻², 0.25 = 2⁻²
Powers of 2 and powers of 3 to know by heart. Fractions and decimals can be powers too: a negative exponent gives a reciprocal.
Worked example

Example 1: Rewrite with base 2

Write each as a power of 2: (a) 8ˣ (b) 4^(x − 1) (c) (1/16)ˣ (d) 32^(x/5) (e) 0.5^(3x)

  1. (a) 8 = 2³: 8ˣ = (2³)ˣ = 2^(3x).
  2. (b) 4 = 2²: 4^(x − 1) = 2^(2(x − 1)) = 2^(2x − 2).
  3. (c) 1/16 = 2⁻⁴: (1/16)ˣ = 2^(−4x).
  4. (d) 32 = 2⁵: 32^(x/5) = 2^(5 × x/5) = 2ˣ. So y = 32^(x/5) is just y = 2ˣ!
  5. (e) 0.5 = 2⁻¹: 0.5^(3x) = 2^(−3x).

2^(3x), 2^(2x − 2), 2^(−4x), 2ˣ and 2^(−3x).

Worked example

Example 2: A translation that is also a stretch

Show that y = 4^(x + 1), y = 4(4ˣ) and y = 2^(2x + 2) are all the same function. Describe it as a transformation of y = 4ˣ in two ways.

  1. Product law: 4^(x + 1) = 4ˣ × 4¹ = 4(4ˣ).
  2. Common base: 4^(x + 1) = (2²)^(x + 1) = 2^(2x + 2).
  3. As y = 4^(x + 1): a translation of y = 4ˣ left 1.
  4. As y = 4(4ˣ): a vertical stretch by a factor of 4. For an exponential graph, these two transformations give the same picture!

All three are equal; the graph is y = 4ˣ shifted left 1, which is the same as stretched vertically by 4.

Worked example

Example 3: Changing to a base that is not a whole-number power

Write 27ˣ as a power of 9.

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

27ˣ = 9^(1.5x)

Worked example

Example 4: Using a common base to solve an equation

Solve (a) 4ˣ = 32 (b) 9^(x − 1) = 27ˣ (c) 2^(x + 1) = 8^(x − 1)

  1. Idea: if two powers of the same base are equal, their exponents are equal (because y = bˣ is always increasing or always decreasing, it never repeats a value).
  2. (a) 2^(2x) = 2⁵, so 2x = 5 and x = 5/2. Check: 4^(2.5) = (√4)⁵ = 32 ✓.
  3. (b) 3^(2(x − 1)) = 3^(3x): 2x − 2 = 3x, so x = −2. Check: 9⁻³ = 1/729 and 27⁻² = 1/729 ✓.
  4. (c) 2^(x + 1) = 2^(3(x − 1)): x + 1 = 3x − 3, so 2x = 4 and x = 2. Check: 2³ = 8 and 8¹ = 8 ✓.

x = 5/2, x = −2 and x = 2.

Looking ahead

When the two sides cannot be written with a common base (for example 2ˣ = 10), you can estimate x from a graph or by guess-and-check for now: 2³ = 8 and 2⁴ = 16, so x is a bit more than 3 (about 3.32). In Grade 12 (MHF4U) you will learn logarithms, which solve such equations exactly.

Just for fun

Some functions go by several names, like a friend who is "Alexander" to teachers, "Alex" at hockey and "Sasha" at home.

9ˣ, 3^(2x) and 81^(x/2) are three names, one function, and exactly one graph.

Hands-on activity

Same graph or not? A calculator investigation

You need: a calculator (or Desmos), paper and a pencil.

  1. Copy this list of pairs: (a) 9ˣ and 3^(2x); (b) 2^(x + 3) and 8(2ˣ); (c) 4^(x/2) and 2ˣ; (d) 2^(3x) and 6ˣ; (e) (1/2)ˣ and 2^(−x); (f) 3^(x + 1) and 3ˣ + 3.
  2. For each pair, predict same or different, then fill a table for x = −1, 0, 0.5, 1, 2.
  3. For each pair that matches, explain it with an exponent law. For each pair that does not, find the first x-value in your table where they differ.
  4. If you have Desmos, graph each pair to confirm.

What to look for: (a), (b), (c) and (e) are the same: (3²)ˣ, 2ˣ × 2³, (2²)^(x/2) and (2⁻¹)ˣ. (d) is different: 2^(3x) = 8ˣ, not 6ˣ (at x = 1: 8 vs 6). (f) is different: 3^(x + 1) = 3 × 3ˣ, which is not 3ˣ + 3 (at x = 0: 3 vs 4; at x = 1: 9 vs 6; the graphs cross only once, between x = 0 and x = 1). Multiplying by a constant and adding a constant are different transformations.

Your turn

1

Write 25ˣ as a power of 5.

2

Write 16ˣ as a power of 2 and as a power of 4.

3

Write (1/27)ˣ as a power of 3.

4

Show that 2^(x + 2) = 4(2ˣ).

5

Describe y = 3^(2x) as a transformation of y = 3ˣ, and name another base for the same function.

6

Solve 2^(3x) = 64.

7

Solve 25^(x + 1) = 125.

More practice: warm-up

1

Write 81 as a power of 3.

2

Write 1/8 as a power of 2.

3

Write 100ˣ as a power of 10.

4

Is 4ˣ equal to 2^(2x)?

5

Solve 3ˣ = 81.

More practice: on your own

1

Write 8^(2x − 1) as a power of 2.

2

Write 125^(x/3) as a power of 5, and describe what this means for the graph.

3

Solve 4^(x − 2) = 8^x.

4

Solve (1/9)ˣ = 27^(x + 5).

5

Show that y = 2^(x − 3) is a vertical compression of y = 2ˣ. By what factor?

Challenge

1

Write 4ˣ as a power of 8.

2

Solve 2^(x²) = 16ˣ.

3

Show that any horizontal translation of y = bˣ can be rewritten as a vertical stretch or compression, and find the factor for a translation right d units.

Check your understanding

Quiz

1.

Which is equal to 9ˣ?

2.

Which is equal to (1/8)ˣ?

3.

y = 2^(x + 3) is the same function as:

4.

Solve 8ˣ = 16.

5.

y = 5^(2x) is y = 5ˣ:

6.

Solve 9^(x + 1) = 27.

For parents and teachers

Big idea: Expectation B2.4: through investigation with technology, see that one exponential function can be written with different bases (e.g., 9ˣ = 3^(2x)) and explain the connection using graphs, transformations and exponent laws.

How to help: Quiz your teen on the powers of 2 (to 2⁶ or 2¹⁰) and of 3 (to 3⁵). Ask them to explain 9ˣ = 3^(2x) three different ways.

Common difficulties:

  • Writing 2^(3x) as 6ˣ: the 3 multiplies the exponent, not the base, so 2^(3x) = (2³)ˣ = 8ˣ.
  • Confusing 3^(x + 1) (= 3 × 3ˣ) with 3ˣ + 3.
  • Forgetting brackets: 4^(x − 1) = 2^(2(x − 1)) = 2^(2x − 2), not 2^(2x − 1).
  • Solving equations when the bases are not the same yet: rewrite both sides with one base first.