About the Logarithm Calculator
What is the Logarithm Calculator?
A logarithm answers the question “to what power must the base be raised to give this number?” Because 10³ = 1,000, log₁₀ 1,000 = 3. Logarithms are the inverse of exponents and turn multiplication into addition.
Base 10 (the common log) is used for pH, decibels and the Richter scale; base e ≈ 2.71828 (the natural log, ln) appears in continuous growth, compound interest and calculus; base 2 is used in computing and information theory. Choose one of these bases or enter your own.
How to use the Logarithm Calculator
- Number (x > 0) — must be greater than 0. (sample: 1,000)
- Base (sample: Base 10 (common log))
- Custom Base (used when "Custom base" is selected) — used only when “Custom base” is selected; must be positive and not 1. (sample: 5)
- Press Calculate Logarithm (results also refresh as you type). The result card shows the Logarithm, along with Natural log (ln x), Common log (log₁₀ x) and Binary log (log₂ x).
Formula used
log_b(x) = ln(x) ÷ ln(b)
- x = the number (greater than 0)
- b = the base (greater than 0 and not equal to 1)
This change-of-base rule lets any logarithm be computed from natural logs. log_b(x) = y means exactly the same as b^y = x.
Worked example
These are the sample values already filled in the calculator above, and the result it returns for them:
| Input | Sample value |
|---|---|
| Number (x > 0) | 1,000 |
| Base | Base 10 (common log) |
| Custom Base (used when "Custom base" is selected) | 5 |
| Result | Value |
|---|---|
| log base 10 of x | 3 |
| Natural log (ln x) | 6.90775528 |
| Common log (log₁₀ x) | 3 |
| Binary log (log₂ x) | 9.96578428 |
Laws of logarithms
- log(a × b) = log a + log b
- log(a ÷ b) = log a − log b
- log(aⁿ) = n × log a
- log_b(b) = 1 and log_b(1) = 0
Tips for using the Logarithm Calculator
- The log of a number between 0 and 1 is negative, and the log of 1 is 0 in every base.
- To solve 2ˣ = 100, take logs of both sides: x = log 100 ÷ log 2 ≈ 6.644.
- Doubling time at r% growth is ln 2 ÷ ln(1 + r/100) — the idea behind the Rule of 72.