What is the change-of-base formula?
The change-of-base formula rewrites a logarithm in one base as a quotient of logarithms in another base. It is especially useful when a calculator directly supports common logarithms and natural logarithms but not the original base.
Here, x is the logarithm argument, b is the original base, and c is the new base. The argument must be positive, and both bases must be positive and different from 1.
How to use this calculator
- Enter the number whose logarithm you want to calculate.
- Enter the original logarithm base.
- Choose natural logarithms, common logarithms, or a custom base.
- If you choose a custom base, enter a positive base other than 1.
- Choose the desired decimal precision and click Calculate logarithm.
Example: evaluate log₂(10)
To find log base 2 of 10, use the natural-log version of the formula:
This means that 2 raised to approximately 3.321928 equals 10.
Why can the conversion base change?
Any positive conversion base other than 1 can be used. Choosing base 10 or base e is convenient because common and natural logarithms are widely available on scientific calculators. A custom base works using the same formula and gives the same mathematical result, apart from rounding.
Frequently asked questions
Does the change-of-base formula change the answer?
No. It rewrites the logarithm in a different form. Any valid conversion base gives the same value, subject to numerical rounding.
Can the argument be zero or negative?
Not for real-valued logarithms. The argument must be strictly greater than zero.
Can the logarithm base be 1?
No. A logarithm base must be positive and different from 1.
What is the difference between log and ln?
In this calculator, log denotes the common logarithm with base 10, while ln denotes the natural logarithm with base e.