Logarithm Calculator
Find the logarithm of a number in any base, plus its natural, common and binary logs.
About the Logarithm Calculator
A logarithm answers: what power must the base be raised to, to get this number? log₁₀(1000) = 3 because 10³ = 1000. The natural log (ln) uses the base e ≈ 2.71828 and appears throughout science and finance.
The formula
log_b(x) = ln(x) ÷ ln(b)
- b
- base: positive, not 1
- x
- number: greater than 0
Log rules
| Rule | Example |
|---|---|
| log(xy) = log x + log y | log 20 = log 2 + log 10 |
| log(x/y) = log x − log y | |
| log(xⁿ) = n·log x | log 1000 = 3·log 10 = 3 |
| log_b(b) = 1 | ln(e) = 1 |
Worked example
Example: log_2(256) = 8
- Inputs: Number (x): 256; Base (b): 2.
- Formula: log_b(x) = ln(x) ÷ ln(b)
- Change of base log_2(256) = ln(256) ÷ ln(2) = 5.5451774445 ÷ 0.6931471806
- Result: log_2(256) = 8
- Natural log ln(x): 5.54517744448
- Common log log₁₀(x): 2.408239965312
- Binary log log₂(x): 8
Questions people ask
What is log of 0?
Undefined; it tends to −∞ as x approaches 0.
Why use ln?
Its rate of change is 1/x, which makes growth and decay formulas neat: continuous interest uses e^(rt).