Log Base 10 Calculator
The common logarithm — how many times you multiply 10 by itself to reach your number.
log₁₀(x) answers "10 to what power gives x?".
How the log base 10 calculator works
log₁₀(x) answers "10 to what power gives x?". Because it counts powers of ten it also counts digits: any number between 1,000 and 9,999 has a log between 3 and 4. That is why it underpins scales that span enormous ranges — the pH scale, the decibel and the Richter magnitude are all base-10 logarithms, so each whole step is a tenfold change.
Formula: log_b(x) = ln(x) / ln(b)
Worked examples
| Inputs | log base b of x | Note |
|---|---|---|
| log 1000 | 3 | 3, because 10³ = 1000 |
| log 2 | 0.3010299957 | 0.30103 |
| log 1,000,000 | 6 | 6 |
FAQFrequently asked questions
What is log base 10?
The power you raise 10 to in order to get your number: log 1000 = 3.
Why is it called the common logarithm?
Because before calculators it was the one printed in tables and used for hand multiplication.
What is log of a negative number?
Undefined in the real numbers — no power of 10 is negative.
Where is it used?
pH, decibels and earthquake magnitude are all base-10 logs, so a step of one means ten times.
How does it relate to digits?
The whole part of the log is one less than the number of digits, which is how slide rules worked.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.