# 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}√e

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

Example, e^{(1/x)}=^{x}√e

**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 π, φ and γ 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*)

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

## 3. EXP Function Example 1

**How to calculate the natural exponential function for four different values [1, 2, 100, -1]?**

Cell range B17:B20 contains the arguments: 1, 2, 100, -1 respectively.

The formula in cell C17:

- Cell C17 calculates the natural exponential of 1 which is approx. 2.71828182845905.
- Cell C18 calculates the natural exponential of 2 which is approx. 7.38905609893065.
- Cell C19 calculates the natural exponential of 100 which is approx. 2.68811714181614E+43.
- Cell C17 calculates the natural exponential of -1 which is approx. 0.367879441171442.

The chart above shows the curves of

- e
^{x}- the blue thicker line. Find values -1, 1, and 2 on the x-axis. From those points, draw imaginary vertical lines upwards until it intersects with the blue curve, which represents the natural exponential function e. Then, follow the points of intersection horizontally towards the y-axis to the left. You will find that the corresponding values on the y-axis is approximately 2.71, 7.38, and 0.37. - 2
^{x }- This curve represents the exponential function with base 2. - 10
^{x}- This curve represents the exponential function with base 10.

The graph displays three exponential curves, the exponential function e^x has a smoother, gradually increasing curve, while 10^x has a much steeper, faster growth rate as x increases. The 2^x curve has a slower growth rate compared to e^{x} and 10^{x}.

## 4. EXP Function Example 2

**In a manufacturing process, the diameter of a particular component follows a normal distribution with a mean of 20 mm and a standard deviation of 0.4 mm. What is the probability density function for a value of 19.8 mm?**

The natural number e is used in the math formula for calculating the probability density function of a normal distribution which is a continuous probability distribution that is widely used in statistics and probability theory.

The text representation of this formula is:

f(x | μ, σ^2) = (1 / (2πσ^2)^0.5) * e^(-(x - μ)^2 / (2σ^2))

- x denotes the probability density function of the normal distribution for a given value of x
- μ represents the mean or expected value of the distribution.
- σ^2 (sigma squared) represents the variance of the distribution which is the square of the standard deviation (σ).

The arguments are:

- x = 19.8 mm
- mean = 20 mm
- standard_dev = 0.4

These arguments are specified in cells C17,C18, and C19 respectively, in the image above.

Formula in cell C22:

This formula in cell C22 is built based on the math formula for calculating the probability density function of a normal distribution described above. It returns 0.88016331691075 which can be found on the y-axis in the chart above.

In the image above, locate the value 19.8 on the x-axis. From that point, draw an imaginary vertical line upwards until it intersects with the blue curve, which represents the probability density function. Then, follow the point of intersection horizontally towards the y-axis to the left. You will find that the corresponding value on the y-axis is approximately 0.88.

### Functions in 'Math and trigonometry' category

The EXP function function is one of 61 functions in the 'Math and trigonometry' category.

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