📈 Logarithm Calculator
Calculate the logarithm of any positive number in base 10, e, 2 or a base of your choice, with the change-of-base working.
Quick answer: log₁₀(1000) = 3 because 10³ = 1000. Its natural log is ln(1000) ≈ 6.907755279, and any base works with the change-of-base rule logᵦ(x) = ln(x) ÷ ln(b).
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Logarithm Calculator inputs
Result
log₁₀(1,000)
= 3
because 10^3 = 1,000
- log₁₀(x)
- 3
- ln(x)
- 6.907755279
- log₂(x)
- 9.9657842847
Step by step
- Definition: log₁₀(1,000) = y means 10^y = 1,000
- Change of base: log₁₀(1,000) = ln(1,000) ÷ ln(10) = 6.907755279 ÷ 2.302585093 = 3
- Check: 10^3 = 1,000
What a logarithm means
A logarithm is the inverse of an exponent. If b raised to the power y equals x, then y is the logarithm of x to base b:
logᵦ(x) = y ⇔ bʸ = x · logᵦ(x) = ln(x) ÷ ln(b)
The second formula, the change-of-base rule, lets you compute a logarithm in any base using only natural logs (or common logs).
The three common bases
| Base | Notation | Typical use |
|---|---|---|
| 10 | log or log₁₀ | pH, decibels, Richter scale, orders of magnitude |
| e ≈ 2.71828 | ln | continuous growth, calculus, compound interest |
| 2 | log₂ or lb | computer science, bits, halving and doubling |
Worked example
log₁₀(1000) = 3, because 10 × 10 × 10 = 1000. For a custom base, take log₃(81): ln 81 ≈ 4.394449 and ln 3 ≈ 1.098612; dividing gives exactly 4, and indeed 3⁴ = 81.
Laws of logarithms
- Product: log(ab) = log a + log b
- Quotient: log(a ÷ b) = log a − log b
- Power: log(aⁿ) = n × log a
- Special values: logᵦ(1) = 0 and logᵦ(b) = 1 for every valid base
Tips
- The number must be greater than 0, and the base must be positive and not equal to 1.
- Logs of numbers between 0 and 1 are negative: log₁₀(0.01) = −2.
- To solve bˣ = y for x, take logs: x = ln(y) ÷ ln(b) – handy for “how long until my money doubles?” questions.
- Each step of 1 in a base-10 log means a factor of 10, which is why an earthquake of magnitude 6 has ten times the wave amplitude of a magnitude 5.
Frequently asked questions
What is a logarithm?
A logarithm answers “what power do I raise the base to, to get this number?”. log₂(8) = 3 because 2³ = 8. Logarithms turn multiplication into addition, which is why they appear in science, finance and computing.
What is the difference between log and ln?
“log” usually means base 10 (the common logarithm), while “ln” is the natural logarithm with base e ≈ 2.718281828. In many programming languages and higher maths texts, however, log means ln – check the context.
How do I calculate a log with a different base?
Use the change-of-base formula: logᵦ(x) = ln(x) ÷ ln(b) (or log(x) ÷ log(b)). For example log₃(81) = ln 81 ÷ ln 3 = 4.394449 ÷ 1.098612 = 4.
Why can’t I take the log of zero or a negative number?
A positive base raised to any real power is always positive, so no exponent produces 0 or a negative number. The logarithm is undefined there; as x approaches 0 the log heads towards minus infinity.