How to use the LOG function
What is the LOG function?
The LOG function calculates the logarithm of a number to a specific base. The log function uses base 10 if a base is not specified.
What is a logarithm?
A logarithm is a mathematical operation that tells you how many times you need to multiply a certain number, called the base, by itself to get another number.
For example, if you want to know how many times you need to multiply 2 by itself to get 16: 2x = 16
you can use a logarithm with base 2 and write it as log2(16) which equals 4.
Now we know the exponent: 24 = 16 or 2 * 2 * 2 * 2 = 16
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.
What is an exponent?
An exponent is a number that indicates the power or times that a base number is multiplied by itself.
Exponents are written as a small raised number, like x2, y3, or 104. The base number is being multiplied by itself as many times as the exponent value. Exponents are also referred to as "raising to a power".
What is the definition of a logarithm with an arbitrary base?
A logarithm with an arbitrary base b can also be defined using a limit definition, similar to how the natural logarithm is defined:
For a base b > 0, b ≠1:
logb(x) = lim (n->infinity) (x^(1/n) - 1) * n / logb(e)
Where:
- b is the base
- x is the input number
- e is the mathematical constant
Here is an example:
b = 3
x = 100
e = 2.718281828
n = large number, here I use 99999999
(100^(1/99999999) - 1) * 99999999/ log3(e) = 4.191806641
=LOG(100,3) returns 4.191806549
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) |
Excel Function Syntax
LOG(number, [base])
Arguments
| number | Required. The positive numerical value for which you want the logarithm. |
| [base] | Optional. The base of the logarithm. Default value is 10. |
Formula in cell C3:
ghfhh
Useful links
Logarithm - wikipedia
Euler's Number - UTSA
Functions in 'Math and trigonometry' category
The LOG function function is one of many functions in the 'Math and trigonometry' category.
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