Logarithm Calculator
This logarithm calculator works for any base: enter the number and the base and the result appears at once, with a check and the change-of-base working. log₂(8) = 3 because 2³ = 8; the ln, lg and lb options pick the standard bases e, 10 and 2.
Example: Logarithm: Any base b: log_b(x) · Base b 2 · Number x 8 → Result: log₂(8) = 3. Source: Wikipedia – Logarithmus (Definition, Rechenregeln, Basiswechsel). Updated: .
How it is calculated
The logarithm logb(x) answers the question: to what power must the base b be raised to get x? Since 2³ = 8, log₂(8) = 3; since 10² = 100, log₁₀(100) = 2. It is the inverse of exponentiation and is defined only for positive x and for bases above 0 other than 1.
How the tool works
- Change of base: Most calculators only have ln and log10. Every other logarithm follows from logb(x) = ln(x) ÷ ln(b). The working shows both intermediate values.
- Check: The base raised to the result must give x again, so a typing slip shows up immediately.
- Estimate: When the result is not a whole number, the tool names the two powers that x lies between. That lets you estimate a logarithm by hand: log₂(50) is between 5 (2⁵ = 32) and 6 (2⁶ = 64).
The main rules
log(a · b) = log(a) + log(b), log(a ÷ b) = log(a) − log(b) and log(aⁿ) = n · log(a). Also logb(1) = 0 and logb(b) = 1. Logarithms answer questions such as how long a doubling takes at a given growth rate. The reverse task is covered by the exponent calculator, and the graphing calculator draws the curve y = log(x).
Frequently asked questions
What is a logarithm in simple terms?
A logarithm is the exponent you need to raise the base to in order to get a number. log₂(16) = 4 because 2 to the power 4 is 16.
How do I calculate a logarithm with any base on a calculator?
Use the change-of-base rule: log_b(x) = ln(x) ÷ ln(b). For log₃(81) enter ln(81) ÷ ln(3) = 4.
What is the difference between ln, log10 and log2?
ln is the natural logarithm with base e ≈ 2.71828, log10 is the common logarithm with base 10 and log2 is the binary logarithm with base 2. All three are special cases of log_b.
Why is there no logarithm of 0 or of a negative number?
A positive base raised to any power never gives 0 or a negative value, so log(x) exists only for x > 0.
Can I work out a logarithm in my head?
Find the powers of the base that bracket the number: 2⁵ = 32 and 2⁶ = 64, so log₂(50) lies between 5 and 6. The calculator shows this bracket as well.
Sources and legal basis
- Wikipedia – Logarithmus (Definition, Rechenregeln, Basiswechsel)
- Wikipedia – Logarithm (change of base)
As of: · Editorial policy
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