Compute logarithms in any base — common log (log₁₀), natural log (ln), binary log (log₂), or any custom base — with full change-of-base working.
Results update live as you type.
log₁₀(1000)
log₁₀(x)
3
common log
ln(x)
6.9078
natural log (base e)
log₂(x)
9.9658
binary log
log_b(x)
6.288
custom base b = 3
| Identity | Form | Example |
|---|---|---|
| Product | log(xy) = log x + log y | log(6) = log 2 + log 3 |
| Quotient | log(x/y) = log x − log y | log(5/2) = log 5 − log 2 |
| Power | log(xⁿ) = n · log x | log(2⁵) = 5 log 2 |
| Identity | log_b(b) = 1 | log₁₀(10) = 1 |
| Zero | log_b(1) = 0 | ln(1) = 0 |
| Change of base | log_b(x) = ln x / ln b | log₂(8) = ln 8 / ln 2 = 3 |
The Definition
A logarithm is the inverse of exponentiation. The statement log_b(x) = y is exactly the same as b^y = x. So log₁₀(1000) = 3 because 10³ = 1000. Calculators normally only have log₁₀ and ln built in — to compute a log in any other base you use the change-of-base formula: log_b(x) = ln(x) / ln(b). Logs have neat properties that turn multiplication into addition (the original purpose of slide rules and log tables) and turn powers into multiplication.
Working for your inputs
About This Tool
A log calculator computes logarithms in any base. The three logs that show up in everyday work are the common log (base 10, written log or log₁₀), the natural log (base e ≈ 2.71828, written ln), and the binary log (base 2, written log₂ or lb). This tool returns all three at once, plus your custom base result, every time you change the argument.
Logarithms were invented by John Napier in 1614 as a way to turn the painful multiplication of long decimal numbers into much easier addition: log(a · b) = log(a) + log(b). For 350 years scientists and engineers carried log tables and slide rules everywhere. Today the practical use has shifted — logs underpin the decibel scale for sound, the Richter scale for earthquakes, the pH scale in chemistry, information theory (entropy in bits = log₂), and algorithm analysis in computer science (O(log n) is binary search; O(n log n) is mergesort).
Mathematically, this calculator uses JavaScript's Math.log (natural log) and Math.log10/Math.log2 for the canonical bases, with the change-of-base formula log_b(x) = ln(x) / ln(b) for any custom base. Results are exact up to IEEE 754 double precision — about 15–17 significant digits.
Use this free log calculator for homework, physics, chemistry, finance (doubling time = ln 2 / r), computer science, or any time you need to invert an exponential. Everything runs in your browser — no sign-up, no tracking, no data leaves your device.
Any Base
log₁₀, ln, log₂ presets plus any custom base you supply.
All Three at Once
Common, natural, and binary log shown side by side.
Inverse Check
Confirms b^y = x — a built-in sanity verification.
Change-of-Base Working
Step-by-step ln(x) / ln(b) so you can show your work.
100% Free & Private
No account, no tracking — every calculation runs locally in your browser.
Curve Visualisation
Plot of y = log_b(x) with your specific point marked.
Pick a base, type the argument, read the answer.
Choose log₁₀, ln, log₂, or Custom. The first three are by far the most common — pick one before reaching for a custom base.
Type the number you want the log of, into x. The argument must be strictly positive — there is no real logarithm of zero or negative numbers.
If you picked "Custom", type the base b. Any positive number except 1 works. Common choices outside the presets: log₃ for ternary, log₁₆ for hex digits.
The summary card shows log_b(x) in your chosen base. Stat tiles show all three standard bases — useful for cross-checking when a textbook asks for one specifically.
The chart shows y = log_b(x) with your point marked. Note how the curve rises steeply near x = 0 and then flattens — logs grow slowly.
The formula card shows the change-of-base arithmetic and verifies b^y = x — exactly the steps you would write out in a homework solution.
Everything you need to know about logarithms — bases, identities, and where they appear.
A logarithm answers the question "what power do I raise the base to in order to get this number?". log_b(x) = y means b^y = x. So log₁₀(1000) = 3 because 10³ = 1000, and log₂(32) = 5 because 2⁵ = 32. Logs are the inverse of exponentials.
By convention, log (with no base shown) means log₁₀ in engineering and elementary maths, and ln means the natural logarithm with base e ≈ 2.71828. In pure mathematics and physics, "log" sometimes means ln — so always check context. In computer science, "log" often means log₂. This calculator shows all three together to avoid ambiguity.
To compute log_b(x) using only logs your calculator natively supports (typically log₁₀ and ln): log_b(x) = log_k(x) / log_k(b) for any valid base k. Using ln: log_b(x) = ln(x) / ln(b). This is how this tool computes custom-base logs internally.
No — the real logarithm log_b(x) is only defined for x > 0. As x → 0⁺, log x → −∞; the log of zero or any negative number is undefined in the real numbers. Complex logarithms extend the definition (log(−1) = iπ for ln) but most calculators, including this one, work in the reals.
log_b(1) = 0 for every valid base b, because b⁰ = 1. Similarly log_b(b) = 1 (since b¹ = b), and log_b(b^n) = n. These three identities — log(1) = 0, log(b) = 1, log(b^n) = n — are the easiest sanity checks when working with logs.
The base e ≈ 2.71828 is "natural" because the derivative of ln(x) is simply 1/x — the cleanest derivative of any logarithm. It also arises naturally in continuous growth: anything compounding at rate r over time t becomes e^(rt). Base 10 is a convention from our number system; base 2 from binary; but base e is the one that falls out of calculus.
Almost everywhere quantities span many orders of magnitude. pH = −log₁₀[H⁺] (chemistry). Decibels use 10·log₁₀ of a power ratio (sound, signal). The Richter scale for earthquakes is log₁₀ of seismic wave amplitude (each "magnitude" is 10× larger). Information in bits = log₂(possibilities). Algorithm analysis — binary search runs in O(log n) time. Compound interest doubling time ≈ ln(2)/r.
The three "laws of logs": log(xy) = log x + log y (product rule), log(x/y) = log x − log y (quotient rule), and log(xⁿ) = n · log x (power rule). These are why logs were so useful before calculators — they turn multiplication into addition, division into subtraction, and powers into multiplication.
Most logs are irrational and have non-terminating decimal expansions — for example log₁₀(2) ≈ 0.30103…. Only special inputs (like x being a power of b) give integer or clean rational answers. The calculator reports 4–8 decimal places, accurate to IEEE 754 double precision.
They are inverse functions: log_b(b^x) = x and b^(log_b x) = x. Graphically, y = log_b(x) is the reflection of y = b^x across the line y = x. Anywhere you see a log, you can rewrite it as an exponential, and vice versa — useful when solving equations involving either.
Yes — any positive b ≠ 1 is a valid base. For 0 < b < 1, log_b(x) is a decreasing function: log_b(1) = 0, but log_b(x) becomes negative for x > 1. The case b = 1 is excluded because 1^y = 1 for all y, so log₁ has no inverse.
Calculations use JavaScript's Math.log / Math.log10 / Math.log2, which are implemented in IEEE 754 double precision — accurate to about 15–17 significant digits. For typical use this is far more precision than any physical measurement could justify. For arbitrary-precision computation you would need a big-decimal library.