Logarithm Calculator
A logarithm is the inverse of exponentiation. If b raised to the power of x equals n (b^x = n), then the logarithm base b of n equals x (log_b(n) = x). This calculator computes the logarithm of any positive number to any base, and automatically displays log base 10 (common logarithm), log base 2, and natural logarithm (ln) for comparison. For example, log₁₀(100) = 2 because 10^2 = 100. The natural logarithm uses e (approximately 2.718) as its base and appears frequently in mathematics, physics, and engineering. Logarithms are essential for solving exponential equations, understanding growth rates, calculating decibels in acoustics, and many other scientific and engineering applications. The change of base formula allows conversion between any logarithm bases.
Logarithm formula
If b^x = n, then log_b(n) = x
Change of base: log_b(x) = ln(x) / ln(b)
Common logarithm: log₁₀(x) = log(x)
Natural logarithm: ln(x) = log_e(x)
Common logarithm values
| Number | Log₁₀ | Ln | Log₂ |
|---|---|---|---|
| 1 | 0.00 | 0.00 | 0.00 |
| 2 | 0.30 | 0.69 | 1.00 |
| 10 | 1.00 | 2.30 | 3.32 |
| 100 | 2.00 | 4.61 | 6.64 |
| 1,000 | 3.00 | 6.91 | 9.97 |
| e | 0.43 | 1.00 | 1.44 |
| π | 0.50 | 1.14 | 1.65 |
Logarithm calculator: frequently asked questions
What is a logarithm?
A logarithm is the inverse of an exponent. If b^x = n, then log_b(n) = x. For example, 10^2 = 100, so log₁₀(100) = 2. Logarithms ask the question: what exponent do I need to raise the base to in order to get this number?
What is the difference between log, log base 10, and natural log?
Log base 10 (written log or log₁₀) uses 10 as the base. Natural log (written ln) uses e (approximately 2.718) as the base. In computer science, log often refers to log base 2. This calculator shows all three for comparison.
What does e mean?
e is Euler's number, approximately 2.718281828. It is a mathematical constant that appears frequently in calculus, exponential growth, and continuous compounding. The natural logarithm (ln) uses e as its base.
Can you take the logarithm of zero or negative numbers?
No, the logarithm is undefined for zero and negative numbers in the real number system. The logarithm is only defined for positive numbers greater than zero.
How do you change the base of a logarithm?
To convert between logarithm bases, use the change of base formula: log_b(x) = log_a(x) / log_a(b). For example, log₂(8) = log₁₀(8) / log₁₀(2).
Official sources
- Logarithms: NIST Special Publication 330.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.