Quick Context
In y = b^x the variable sits in the exponent. Each step of 1 in x multiplies y by b, rather than adding to it. That single difference is what separates exponential from linear behaviour.
Three Regimes, Set by the Base
- b > 1 — growth. Each step multiplies up. Compound interest, viral spread, unchecked model divergence.
- b = 1 — flat. 1 to any power is 1. The boundary between the other two.
- 0 < b < 1 — decay. Each step multiplies down, approaching zero without ever arriving. Learning-rate schedules, radioactive decay, the vanishing gradient.
Slide the base across 1 and watch the curve flip from climbing to falling.
Exponential Always Wins Eventually
Keep the polynomial comparison on and look at the left of the plot: x³ is far ahead. Now slide x rightward. There is a crossover point, and after it 2^x leaves x³ hopelessly behind.
This holds for any polynomial and any base above 1 — exponential growth always overtakes eventually. It is the reason exponential-time algorithms are considered intractable no matter how fast the hardware, and the reason a "small" compounding effect is never actually small.
Doubling Time and Half-Life
doubling time = log(2) / log(b)
half-life = log(0.5) / log(b)
A defining feature of exponentials: the time to double is constant. It takes just as long to go from 1 to 2 as from a million to two million. The panel computes this live — at base 2 it is exactly 1 step, at base 1.1 about 7.3 steps.
Why e Is Special
Set the base to e ≈ 2.71828. This is the base at which the curve's slope equals its own height at every point:
d/dx (e^x) = e^x
No other base has this property, which is why e turns up throughout calculus and why exp is the exponential in almost every ML formula — softmax, sigmoid, Gaussian densities and exponential learning-rate decay all use it.
Interactive Exploration Guide
- Base 2, slide x from 0 to 10. The values are 1, 2, 4, 8… 1024. The first few steps look tame; the last is anything but.
- Watch the crossover with x³. Note how far right it happens — exponential growth hides for a surprisingly long time before it dominates.
- Set the base to 0.5. Now it is halving: 1, 0.5, 0.25… approaching zero but never reaching it. This is exactly a learning-rate decay schedule.
- Tick log-scale y. The exponential becomes a perfectly straight line — that is the signature of exponential data, and how you spot it in a real chart.
- Try base 1.05 (5% growth). It looks nearly flat, yet the doubling-time panel says about 14 steps — compound interest in one number.
Key Takeaway
Exponentials multiply rather than add, so they have a constant doubling time and eventually beat any polynomial. Bases above 1 explode, bases below 1 decay toward zero, and base e is the one whose slope equals its value — which is why it underlies softmax, sigmoid and every decay schedule you will configure.