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 102Â = 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)1Â = 2
(1 + 1/2)2Â = 2.25
(1 + 1/3)3Â = 2.37037
The limit of this exponential growth pattern is the unique number e.
e = e1Â equals 2.71828182845904.
e2Â 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. LN Function Syntax
LN(number)
3. LN Function Arguments
number | Required. The positive numerical value for which you want the natural logarithm. |
4. LN Function Example
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:
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 73 functions in the 'Math and trigonometry' category.
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