Logarithm Calculator
Free logarithm calculator. Compute log base b of any number, plus natural log and base-10 log. Works with any base. No signup.
What a logarithm calculator does
A logarithm answers one question: to what power must the base be raised to produce this number? The logarithm of 1000 to base 10 is 3, because 10³ = 1000. This calculator works with any valid base and also reports the natural log, the base-10 log, and an antilog check that raises the base back to the result.
How the calculation works
Browsers do not offer an arbitrary-base log function, so the tool uses the change-of-base formula:
log_b(x) = ln(x) ÷ ln(b)
It computes the natural logarithm of the number and of the base, then divides them. Alongside that it shows ln(x) directly and the base-10 log, and finally raises the base to the computed logarithm to prove the answer: b raised to log_b(x) returns x. The argument must be positive, the base must be positive, and the base cannot be 1. Results are shown to ten significant digits.
Worked example using the default values
The form loads number = 1000 and base = 10, so the tool evaluates log₁₀(1000).
| Quantity | Working | Result |
|---|---|---|
| log₁₀(1000) | ln(1000) ÷ ln(10) = 6.907755279 ÷ 2.302585093 | 3 |
| Natural log ln(1000) | ln(1000) directly | 6.907755279 |
| Base-10 log | log₁₀(1000) | 3 |
| Antilog check | 103 | 1000 |
The antilog check equals the input, so the logarithm is correct. The ratio ln(1000) ÷ ln(10) is about 3 as expected because 1000 is exactly 10 cubed.
Reference: useful logarithms
| Expression | Value | Because |
|---|---|---|
| log₁₀(100) | 2 | 10² = 100 |
| log₁₀(0.01) | −2 | 10−2 = 0.01 |
| log₂(8) | 3 | 2³ = 8 |
| log₂(1024) | 10 | 210 = 1024 |
| log₅(625) | 4 | 5⁴ = 625 |
| ln(1) | 0 | e⁰ = 1 |
| ln(e) | 1 | e¹ = e |
The three log laws let you break products and powers apart: log(xy) = log(x) + log(y), log(x ÷ y) = log(x) − log(y), and log(xn) = n × log(x). These rules are why logarithms turn multiplication into addition.
Common mistakes
- Taking the log of zero or a negative number. No power of a positive base produces zero or a negative result, so the tool refuses these inputs.
- Choosing base 1. Every power of 1 equals 1, so log base 1 has no single answer. The tool ignores a base of 1.
- Confusing ln with log₁₀. They differ by the constant ln(10) ≈ 2.302585, so swapping them scales the answer by that factor.
- Reading a negative log as an error. A result below zero simply means the number is below 1, as with log₁₀(0.01) = −2.
- Forgetting the base is not always 10. The default is 10, but engineering work often uses base 2 or the natural base e.
What is an antilog and how do I use it?
An antilog reverses a logarithm: it raises the base to the log value. If log₁₀(x) = 3, the antilog is 10³ = 1000. The tool performs this automatically as a check row, so you can confirm the logarithm without a second calculation.
Which base should I use?
Use base 10 for scales built on powers of ten such as pH, decibels, and earthquake magnitude. Use base 2 for anything binary, including bit depth, binary search depth, and information content. Use the natural base e for continuous growth and decay, compound interest, and half-life problems.
Why is the natural log so common in mathematics?
Because the function ex is its own derivative, which makes e the natural base for calculus and for models of continuous change. Any other base would introduce a constant multiplier into every derivative involving a logarithm.
Can the logarithm be a fraction?
Yes. Any real number is a possible logarithm, positive or negative. log₂(3) is approximately 1.584963, and log₁₀(3) is approximately 0.477121. Whole-number answers occur only when the argument is an exact power of the base.
The sign of the result is a useful check: a positive logarithm means the argument is greater than one, and a negative logarithm means it sits between zero and one. Extreme magnitudes are often easier to handle in exponent form, so the scientific notation converter is a natural companion to this tool.