# How to use the EXP function

**What is the EXP function?**

The EXP function returns e raised to the power of a given number.

### Table of Contents

## 1. Introduction

**What is e?**

It is the base of the natural logarithm, ln.

The chart above shows the curve of e^{x}, 2^{x}, and 10^{x}

**What is the definition of e?**

e is defined to be the limit of (1 + 1/n) raised to the nth power as n approaches infinity.

In equation form: e = lim (1 + 1/n)^{n} as n -> infinity

The limit of this exponential growth pattern is the unique number e. For example:

(1 + 1/1)^{1} = 2

(1 + 1/2)^{2} = 2.25

(1 + 1/3)^{3} = 2.37037

The limit of this exponential growth pattern is the unique number e.

e = e^{1} equals approx. 2.71828182845904.

e^{2} equals approx. 7.389056099

**What is the EXP function in Excel an abbreviation of?**

The "EXP" name refers to Exponent or Exponential.

**What is ln?**

ln is the natural logarithm and e is it's base.

e^{ln x} = x

ln e^{x} = x

**What are the key exponential rules?**

- Product of Powers

e^x * e^y = e^(x+y)

When two numbers with exponents are multiplied together, and the bases are the same, you add the exponents.

Example, e^{5}* e^{10}= e^{(5+10)} - Quotient of Powers

e^x / e^y = e^(x-y)

Two numbers with exponents are divided and the bases are the same you subtract the exponents.

Example, e^{10}/ e^{5}= e^{(10-5)} - Power of Powers

(e^x)^y = e^(x*y)

If a base is raised to a power and the entire expression is further raised to another power then the two powers are multiplied, and the base remains unchanged.

Example, (e^{10})^{5}= e^{(10*5)} - Power of Product

(ey)^x = e^x * y^x

When a product expression within parentheses is raised to an exponent, each element within the product is raised to that exponent individually.

Example, (5*e)^{10}= e^{10}* e^{5} - Negative Exponent

e^-x = 1/e^x

A negative exponent indicates that the base should be taken as the reciprocal and raised to the positive equivalent of the exponent.

Example, e^{-x}= 1/e^{x} - Fractional Exponent

e^(1/x) =^{x}âˆše

A number raised to a fractional exponent is equivalent to taking the root of that number.

Example, e^{(1/x)}=^{x}âˆše

**What applications has e?**

- Logarithms to base e models growth/decay.
- Compound interest - e models exponential growth in finance.
- Probability theory - e features in Poisson distributions.
- Normal distributions - e in its probability density function.
- Euler's identity - e^ix = cos(x) + i*sin(x) links e with trigonometry and the imaginary unit i.
- Bacterial growth - The growth of bacterial populations can be modeled involving e.
- Mathematical constants - e is related to mathematical constants like Ï€, Ï† and Î³ via infinite series and limits.
- Calculus - The derivative of e^x is itself e^x.

**What logarithmic functions exist in Excel?**

Name | Excel Function | Description |
---|---|---|

Natural Logarithm | LOG(x) | Calculates the natural logarithm (base e) of x |

Natural Logarithm | LN(x) | Same as LOG(x), calculates the natural log of x |

Common Logarithm | LOG10(x) | Calculates the base 10 logarithm of x |

Binary Logarithm | LOG2(x) | Calculates the base 2 logarithm of x |

Exponentiation | POWER(x, y) | Raises x to the power of y |

Exponential | EXP(x) | Raises e to the power of x |

## 2. EXP Function Syntax

EXP(*number*)

## 3. EXP Function Arguments

number |
Required. The number used as an exponent to e. |

## 4. EXP Function Example

Formula in cell C3:

Graph of the natural exponential function.

### Functions in 'Math and trigonometry' category

The EXP function function is one of 73 functions in the 'Math and trigonometry' category.

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