# How to use the LN function

**What is the LN function?**

The LN function calculates the natural logarithm of a number.

### Table of Contents

## 1. Introduction

**What is ln?**

Ln is an abbreviation of natural logarithm are based on the constant e. It is the inverse of the exponential function.

**How is the natural logarithm defined?**

The natural logarithm, or log base e, is defined using the following limit:

ln(x) = lim (n->infinity) (x^{(1/n)} - 1) * n

**What are the four key natural logarithm rules?**

The following rules are essential to learn if you want to solve various equations involving natural logarithms effectively.

1. Product rule

ln(x*y) = ln(x) + ln(y)

The natural log of the multiplication of x and y is the sum of the ln x and ln y.

Example, ln (5*10) = ln(5) + ln(10)

2. Quotient rule

ln(x/y) = ln(x) - ln(y)

The natural log of the division of x and y is the difference of the ln x and ln y.

Example, ln (5/10) = ln(5) - ln(10)

3. Reciprocal rule

ln(1/x) = − ln(x)

The natural log of the reciprocal of x is similar to the quotient rule. ln(1/x) = ln(1) - ln(x) = 0 - ln x = - ln x

Example, ln(1/5) = - ln(5)

4. Power rule

ln(x^{y}) = y * ln(x)

The natural log of x raised to the power of y is y multiplied by the ln x.

Example, ln(5^{10}) = 10 * ln(5)

Some key properties of the natural logarithm:

- The natural log of 1 is 0.
- LN is undefined for negative numbers.
- LN is the inverse of the EXP function

LN(EXP(x)) = x.

**What is e?**

E stands for Euler's constant and it is the **base** of the natural logarithm shortened to ln. This means that ln e = 1

Excel has the function named EXP that returns e raised to the power of a given number.

**What is a base?**

In logarithms, the base is the number that is raised to a power to produce the desired output. It is the foundation of the logarithmic function.

For example, the logarithm log10(100):

- 10 is the base
- 100 is the input number
- 2 is the exponent that makes 10 return 100 because 10
^{2}= 100

The most common bases are

- 10
- e (natural log), and
- 2 (for computers).

But any positive number besides 1 can be a base.

**How is e defined?**

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. Here are some examples:

(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 2.71828182845904.

e^{2} equals 7.389056099

**What are the other logarithmic functions in Excel?**

Excel function | Description |
---|---|

LOG | Returns the logarithm with a given base. |

LN | Returns the natural logarithm (base e) of a number |

LOG10 | Returns the base-10 logarithm of a number |

LOG2 | Returns the base-2 logarithm of a number |

**The corresponding inverse functions for the logarithm functions in Excel:**

Logarithm | Power to |
---|---|

LOG | Arbitrary base, base^{x} or POWER(number, power) |

LN | EXP(number) |

LOG10 | 10^{x} or POWER(10,x) |

LOG2 | 2^{x} or POWER(2,x) |

**What applications does the natural logarithm have?**

Natural logs have applications in math, science, and finance for calculating compound growth, diffusion, acidity, and more. LN is useful whenever natural exponential growth or decay is involved.

**What is the difference between the natural logarithm (ln) and the base 10 logarithm (log _{10})?**

The difference between natural logarithm and the base 10 logarithm is the base being used. The natural logarithm uses e as the base whereas the log_{10} uses 10 as the base.

## 2. Syntax

LN(*number*)

number |
Required. The positive numerical value for which you want the natural logarithm. |

## 3. Example 1

This example shows the different logarithms on an Excel chart, the logarithms are log_{2}, ln, and log_{10}

The logarithms return a negative result if the x value is larger than 0 (zero) and smaller than 1. The logarithm for 0 (zero) is undefined resulting in a #NUM error. The logarithms return a positive result if the x value is larger than 1.

x | log_{2} |
ln | log_{10} |
---|---|---|---|

0 | #NUM! | #NUM! | #NUM! |

1 | 0.000 | 0.000 | 0.000 |

2 | 1.000 | 0.693 | 0.301 |

3 | 1.585 | 1.099 | 0.477 |

4 | 2.000 | 1.386 | 0.602 |

5 | 2.322 | 1.609 | 0.699 |

6 | 2.585 | 1.792 | 0.778 |

7 | 2.807 | 1.946 | 0.845 |

8 | 3.000 | 2.079 | 0.903 |

9 | 3.170 | 2.197 | 0.954 |

10 | 3.322 | 2.303 | 1.000 |

The table above is rounded to three decimals.

Formula in cell B3:

The result of ln(1) is 0 (zero), the image above shows a graph containing the natural logarithm. The curve has the color orange, find 1 on the x axis. It intersects the x-axis at 1 which equals 0 (zero) on the y-axis.

The chart also plots the log2 and log10 curves:

- The log2x curve (blue) grows the fastest among the three, meaning for any given x value, log2x has the largest y-value.
- The ln curve (orange) has a slower growth rate than log2x but faster than log10x.
- The log10x curve (grey) grows the slowest, having the smallest y-values for any given x.

All three curves share some common characteristics:

- They are increasing functions, meaning their values increase as x increases along the x-axis.
- They have a vertical asymptote at x = 0, meaning they approach negative infinity as x approaches 0 from the positive side.
- They are concave downwards, with the rate of increase gradually slowing down as x grows larger.

## 4. Example 2

**How many years will it take to compound 3300, based on 5% interest rate, to a future value of 6900?**

The LN function allows us to calculate how many periods ( or years in this example) based on the following data:

- r - interest rate (This value is specified in cell C19)
- PV - present value (This value is found in cell C18)
- FV - future value (This value is in cell C17)

The formula for calculating the number of periods (n) needed is:

n = ln( FV / PV ) / ln(1+r)

Formula in cell C21:

The formula returns 15.1177794886966 periods, lets see if we can calculate this value.

n = ln (6900 / 3300)/ ln(1 + 0.05)

n = 0.737598943130779/0.048790164169432

n = 15.1177794886966

The chart above shows a orange curve representing the the interest rate and a blue curve representing the cumulative compounded value across periods. The data points in the graph are based on data in cell range B25:D41

## 5. How to graph the natural logarithm in Excel?

To create the chart above I started entering x values in one column, to get a finer curve I entered x values with smaller and smaller increments as I got closer to 0 (zero), see the table below.

The next column has the result of the natural logarithm, this will be the y (vertical) values of the line on the chart.

x |
LN |

0.04 | -3.2188758248682 |

0.05 | -2.99573227355399 |

0.06 | -2.81341071676004 |

0.07 | -2.65926003693278 |

0.08 | -2.52572864430826 |

0.09 | -2.40794560865187 |

0.1 | -2.30258509299405 |

0.2 | -1.6094379124341 |

0.3 | -1.20397280432594 |

0.4 | -0.916290731874155 |

0.5 | -0.693147180559945 |

0.6 | -0.510825623765991 |

0.7 | -0.356674943938732 |

0.8 | -0.22314355131421 |

0.9 | -0.105360515657826 |

1 | 0 |

1.1 | 0.0953101798043249 |

1.2 | 0.182321556793955 |

1.3 | 0.262364264467491 |

1.4 | 0.336472236621213 |

1.5 | 0.405465108108164 |

1.6 | 0.470003629245736 |

1.7 | 0.53062825106217 |

1.8 | 0.587786664902119 |

1.9 | 0.641853886172395 |

2 | 0.693147180559945 |

3 | 1.09861228866811 |

4 | 1.38629436111989 |

5 | 1.6094379124341 |

6 | 1.79175946922806 |

7 | 1.94591014905531 |

8 | 2.07944154167984 |

9 | 2.19722457733622 |

10 | 2.30258509299405 |

Select both columns and then go to "Insert" tab on the ribbon.

### Functions in 'Math and trigonometry' category

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

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