What is a logarithm?

A logarithm is the power to which a number must be raised in order to express some other number. For example in the equation 125 = 53, we call 5 the 'base' and 3 the power or index. We can use logarithms to write the equation in another form, such as:

Log5 125 = 3

This is read as "logarithm to the base 5 of 125 is 3".

As a general rule, the logarithmic equation is as follows:

$ \newcommand{\ctext}[1]{\style{font-family:Arial}{\text{#1}}}  \ctext{If } y = a^x,\ctext{ then } log_ay = x,\ctext{ where } a \gt 0, a \ne 1 $

While the base of a logarithm can be any positive number other than 1, the commonly used bases are 10 and e.

Logarithms to base 10 are often denoted by 'log' or 'log10' and are referred to as common logarithms, while logarithms to base e are denoted by 'In' or 'loge' and are usually called natural logarithms.

For example, since 100 = 102, then the log of 100 is 2 because 10 raised to the second power is 100. Therefore log 102 = 2

Another example, since 1,000,000 = 106, the log of 1,000,000 is 6 because 10 raised to the sixth power is 1,000,000. Therefore log 106 = 6

Below is a table outlining some of the most common logarithms of powers of ten.

1,000,000 = 106 = one million

Log 106 = 6

100,000 = 105 = one hundred thousand

Log 105 = 5

10,000 = 104 = ten thousand

Log 104 = 4

1,000 = 103 = one thousand

Log 103 = 3

100 = 102 = one hundred

Log 102 = 2

10 = 101 = ten

Log 101 = 1

1 = 100 = one

Log 100 = 0

0.1 = 10-1 = one tenth

Log 10-1 = -1

0.01 = 10-2 = one hundredth

Log 10-2 = -2

0.001 = 10-3 = one thousandth

Log 10-3 = -3

0.0001 = 10-4 = one ten thousandth

Log 10-4 = -4

0.00001 = 10-5 = one hundred thousandth

Log 10-5 = -5

0.000001 = 10-6 = one millionth

Log 10-6 = -6


A natural logarithm refers to logarithms that have a base of e, which is approximately 2.7183, abbreviated as 'In'. Note that the terms 'log' and 'In' are not synonyms and should never be used interchangeably.

Previous  | Next