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Glacius
FoundationsConcept reference

Exponential functions

An exponential function scales a starting value by a positive constant base raised to the input.

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. Start with 3
  3. A base between 0 and 1 gives decay
  4. Check your understanding
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01Start with 3#

Start with 3. If each round doubles the amount, the values are 3, 6, 12. The change is 3 at first and 6 next; the multiplier stays 2.

The exponential curve 3 times 2 to the x has values 3,6,12 at inputs 0,1,2. Equal horizontal steps multiply the output by 2 rather than adding a constant.The exponential curve 3 times 2 to the x has values 3,6,12 at inputs 0,1,2. Equal horizontal steps multiply the output by 2 rather than adding a constant.
Figure 1The exponential curve 3 times 2 to the x has values 3,6,12 at inputs 0,1,2. Equal horizontal steps multiply the output by 2 rather than adding a constant.
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An exponent counts repeated scaling.

f(n)=3⋅2nf(n)=3\cdot2^n

At n=0n=0, the value is 3. After two steps, f(2)=3⋅2⋅2=12f(2)=3\cdot2\cdot2=12. The rule abnab^n starts at aa and multiplies by bb per step.

Check your reasoning

For f(n)=3⋅2nf(n)=3\cdot2^n, the current value is 12. What is the value one step later?

Show answer and explanation
24

One more input step multiplies the current value by 2: 12×2=2412\times2=24.

02A base between 0 and 1 gives decay#

A base between 0 and 1 gives decay. With base 1/21/2, each step halves the current value: 16, 8, 4. Use a positive base; base 1 would leave the value constant.

Check your reasoning

A value starts at 20 and is multiplied by 1/21/2 at each step. What is it after two steps?

Show answer and explanation
5

First 20×(1/2)=1020\times(1/2)=10, then 10×(1/2)=510\times(1/2)=5.

03Check your understanding#

Check your reasoning

A process has 4 copies and triples its total each round. How many copies after two rounds?

Show answer and explanation
36

Each round triples: 4 becomes 12, then 36.

Key takeaway

For each input step of one, multiply the current output by the base.

  • Predict multiplicative change from an exponential rule

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Exponential functions.” Math behind ML. /learn/f-exponential