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(xy) = y * ln(x)
The natural log of x raised to the power of y is y multiplied by the ln x.
Example, ln(510) = 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 102 = 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 = e1 equals 2.71828182845904.
e2 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, basex or POWER(number, power) |
| LN | EXP(number) |
| LOG10 | 10x or POWER(10,x) |
| LOG2 | 2x 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 (log10)?
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 log10 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 log2, ln, and log10
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 | log2 | ln | log10 |
|---|---|---|---|
| 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 62 functions in the 'Math and trigonometry' category.
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