📈 Logarithm & Natural Log (ln) Calculator
Solve for any part of log₆(x) = y — the logarithm, the antilog or the base — with the change-of-base working shown, in base 10, 2, e or any other.
What Logarithm & Natural Log (ln) Calculator Does
A logarithm answers a single question: what power turns this base into that number? log₁₀(100) is 2 because 10² is 100. That is all it is — the inverse of raising to a power, and every property follows from that one fact.
The reason logarithms earn their keep is that they turn multiplication into addition. Multiplying two numbers becomes adding their logarithms, which is why slide rules worked and why log tables were worth centuries of effort to compute. It is also why logarithmic scales appear wherever quantities span many orders of magnitude — decibels, pH, the Richter scale, stellar magnitudes.
Three bases dominate and they are not interchangeable in practice, though they differ only by a constant factor. Base 10 maps onto orders of magnitude and suits science and engineering. Base e is what calculus produces naturally and describes continuous growth. Base 2 counts doublings, which is the currency of computer science.
The direction people most often need and least often find is the reverse one. Given a base and a result, recovering the original number is the antilogarithm — simply the base raised to that power. Leave the argument blank on this page and it solves that way instead, which is what a log table did when you read it backwards.
How to Use Logarithm & Natural Log (ln) Calculator
- Enter argument value x and base b
- Review computed log_b(x) alongside natural log ln(x) and binary log log₂(x)
- Inspect exponential verification formula check
Formula Used by Logarithm & Natural Log (ln) Calculator
The definition
logᵦ(x) = y ⟺ bʸ = x
- b
- The base — must be positive and not equal to 1
- x
- The argument — must be positive
- y
- The result: the exponent that turns b into x
Worked example
log₁₀(1000)
- Ask: 10 to what power gives 1000?
- 10 × 10 × 10 = 1000, so three tens
Result: 3 — and the check runs the other way: 10³ = 1000
Change of base — how calculators actually do it
logᵦ(x) = ln(x) ÷ ln(b), or logₖ(x) ÷ logₖ(b) for any valid base k
- k
- Any base you can already compute — usually e or 10
Worked example
log₇(100), where no calculator has a base-7 button
- ln(100) = 4.605170
- ln(7) = 1.945910
- 4.605170 ÷ 1.945910
Result: 2.366589 (the rounded intermediates above give 2.366588; carry full precision to land on the last digit)
Antilogarithm — running it backwards
x = bʸ
- Antilog
- The inverse operation: given the base and the logarithm, recover the argument
Worked example
The antilog of 2 in base 10, and of 8 in base 2
- 10² = 100
- 2⁸ = 256
Result: 100 and 256 — the same relationships as log₁₀(100) = 2 and log₂(256) = 8, read the other way
The log rules, with worked values
Every one follows from the definition. These are what turn multiplication into addition.
| Rule | Statement | Worked example |
|---|---|---|
| Product | logᵦ(xy) = logᵦ(x) + logᵦ(y) | log(1 × 10) = 0 + 1 = 1 |
| Quotient | logᵦ(x/y) = logᵦ(x) − logᵦ(y) | log(10 ÷ 2) = 1 − 0.301 = 0.699 |
| Power | logᵦ(xⁿ) = n × logᵦ(x) | log(2⁶) = 6 × 0.301 = 1.806 |
| Change of base | logᵦ(x) = logₖ(x) ÷ logₖ(b) | log₇(100) = ln(100) ÷ ln(7) = 2.3666 |
| Swap base and argument | logᵦ(c) = 1 ÷ log𝒸(b) | log₅(2) = 1 ÷ log₂(5) = 0.430677 |
The three common bases on the same numbers
They differ only by a constant factor — ln(x) is always about 2.303 × log₁₀(x) — but each reads naturally in its own field.
| x | log₁₀(x) | ln(x) | log₂(x) |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 0.3010 | 0.6931 | 1 |
| 10 | 1 | 2.3026 | 3.3219 |
| 100 | 2 | 4.6052 | 6.6439 |
| 256 | 2.4082 | 5.5452 | 8 |
| 1000 | 3 | 6.9078 | 9.9658 |
| e ≈ 2.71828 | 0.4343 | 1 | 1.4427 |
Values worth knowing, and the ones that do not exist
The undefined cases are not a limitation of any calculator — no exponent produces zero or a negative number from a positive base.
| Expression | Value | Why |
|---|---|---|
| logᵦ(1) | 0 | Any base to the power 0 is 1 |
| logᵦ(b) | 1 | The base to the first power is itself |
| ln(eˣ) | x | The two operations cancel exactly |
| logᵦ(0) | Undefined | No exponent drives a positive base to zero |
| logᵦ(negative) | Undefined in the reals | A positive base raised to any real power stays positive |
| log₁(x) | Undefined | 1 to any power is 1, so it can never reach another value |
How to Read Your Result
Why logarithmic scales exist
When data spans many orders of magnitude, a linear axis is useless — plot earthquake energies linearly and every event but the largest collapses onto the baseline. Taking logs compresses the range so that equal distances represent equal ratios rather than equal differences. This is why the Richter scale, decibels, pH and stellar magnitude are all logarithmic, and why a step of one on any of them means a fixed multiple rather than a fixed amount.
Base 2 counts doublings
log₂(x) answers how many times you must double 1 to reach x, which is exactly the question binary search, tree depth and algorithmic complexity ask. log₂(1,000,000) is roughly 20, so a balanced search through a million items takes about twenty comparisons. That is the whole intuition behind why logarithmic algorithms scale so well: multiplying the data by a thousand adds only ten steps.
The natural log is not arbitrary
Base e looks like an odd choice until you differentiate. The derivative of ln(x) is exactly 1/x, with no constant hanging off it, and eˣ is the only function that is its own derivative. Anything growing at a rate proportional to its current size — compound interest, population, radioactive decay — is described by e without needing a correction factor. That is why calculus produces it whether or not you asked for it.
What the change of base rule buys you
A calculator with only ln and log₁₀ can compute a logarithm in any base whatsoever, because logᵦ(x) = ln(x) ÷ ln(b). This is not a workaround, it is how essentially every implementation does it internally, including this one. It also explains why the three common bases differ by constant factors: dividing by ln(10) or ln(2) is just a fixed rescaling.
Reading a logarithm as a magnitude
In base 10 the integer part tells you the number of digits before the decimal point, and the fractional part tells you where within that decade the value sits. log₁₀(5000) is 3.699 — the 3 says it is in the thousands, the .699 says it sits about seven tenths of the way up that decade on a log scale. This is exactly the reading a slide rule exploited, and it is a fast way to sanity-check an order of magnitude in your head.
Limitations & Accuracy Notes
- Only positive arguments and positive bases other than 1 are defined. Logarithms of negative numbers exist in the complex plane but are outside what this tool computes.
- Results are floating-point approximations. Values like log₁₀(1000) come out exact, but most logarithms are irrational and are rounded for display.
- For arguments extremely close to zero the result grows without bound, and precision degrades before the value becomes formally undefined.
- The antilog direction can overflow quickly: a large base with a large exponent exceeds what a double-precision number can represent, and the tool reports no result rather than infinity.
- This computes logarithms of numbers. It does not solve logarithmic equations symbolically — Symbolab and Mathway handle that, and their step-by-step algebra is a different capability from arithmetic evaluation.
Frequently Asked Questions
What is a logarithm?
What is an antilog?
What is the difference between ln, log and log₂?
How does the change of base formula work?
Why can I not take the log of zero or a negative number?
What is the difference between log and ln?
How do I calculate a logarithm in another base?
What happens to a logarithm as the input approaches zero?
What is a logarithm actually used for?
What does it mean that a scale is logarithmic?
Is my data stored?
References & Further Reading
- NIST Digital Library of Mathematical Functions — Elementary functions — US National Institute of Standards and Technology reference for the logarithm, its identities and its domain
- Calculator.net — Log Calculator — The most complete competing rules section; its worked examples were reproduced to cross-check the identities quoted here