Chapter 02

Exponential and Exponential Functions

Exponentiation is repeated multiplication of the same base; an exponential function fixes the base and uses the exponent as the variable. Used in activation and loss design in deep learning.

Math diagram by chapter

Select a chapter to see its diagram below. View the flow of basic math at a glance.

Example: $2^0=1$ , $2^1=2$ , $2^2=4$ , $2^3=8$

What are exponent and exponential function?

An exponent indicates how many times a number is multiplied by itself. Example: 2 multiplied 3 times is

2^3 = 2 \times 2 \times 2 = 8

. Here 3 is the exponent.

An exponential function fixes a base

a

and outputs

a^x

for input

x

. We write

y = a^x

(

a > 0

a \neq 1

). If

a > 1

, the value grows as

x

increases; if

0 < a < 1

, it decreases as

x

increases.

The natural constant $e$ (about 2.718…) is a special base used often in math and deep learning. It appears naturally when describing “natural growth,” and the derivative of

e^x

e^x

itself, which keeps formulas simple. Softmax and probability models use

e^z

for activations and probability computations.

In AI, exponential functions appear in activation functions (e.g.

e^x

inside softmax) and in loss and probability design. With logarithms they turn products into sums and simplify computation.

In deep learning, softmax applies

e^z

(score

z

) to each output to form a probability-like distribution. Without exponents you can't read *why* that computation. Knowing exponent and log helps you understand activation and loss design.

Loss and probability models often use expressions involving exponents. With basic exponent and exponential functions you can follow where and how they are used.

In AI, exponential functions are used as 'input a score (number), output a positive number.' Softmax uses

e^z

on each score so they sum to 1, then picks one. Knowing exponents lets you read this process.

Expression	Value
$2^0$	1
$2^1$	2
$2^2$	4
$2^3$	8
$2^4$	16
$3^2$	9
$3^3$	27

In the visual below,

y = 2^x

gives

1

for

x=0

2

for

x=1

4

for

x=2

8

for

x=3

. Use it to see how base and exponent relate.