How to Use
- Select logarithm type
Choose common log (log base 10), natural log (ln), or custom base.
- Enter the value
Input a positive number. For custom base, also enter the base value.
- View result
Click Calculate to see the logarithm value and related properties.
What is a logarithm?
A logarithm is the inverse of exponentiation. log_b(x) = y means 'raising base b to the power y gives x', which is equivalent to b^y = x. For example, multiplying 10 by itself twice gives 100, so log₁₀(100) = 2.
Logarithms matter because they turn multiplication into addition. They let you compare very large and very small numbers at a glance, which is ideal for values that span many orders of magnitude.
- Earthquake magnitude the Richter scale is based on the common logarithm
- Sound intensity decibels (dB)
- Acidity pH = -log₁₀[H⁺]
- Algorithmic complexity O(log n)
This calculator lets you set the base freely and at the same time shows the natural, common, and binary logarithm values together.
Calculation formula
This calculator uses the change of base formula to compute a logarithm for any base.
log_b(x) = ln(x) / ln(b)
- x: the positive number whose logarithm you want (the argument)
- b: the base of the logarithm (b > 0, b ≠ 1)
- ln: the natural logarithm, with the constant e (≈2.71828) as its base
Example to find log₂(8): ln(8) / ln(2) = 2.0794 / 0.6931 = 3. Indeed 2³ = 8, so the result matches. Results are shown to 10 decimal places.