Log Calculator
Calculate log base 10, natural log (ln), log base 2, any custom base logarithm, and antilog — with complete step-by-step solutions and a reference table.
Common logarithm reference values for log₁₀, ln, and log₂:
| x | log₁₀(x) | ln(x) | log₂(x) |
|---|
f Formulas Used
? How To Use
★ Key Features
6 Log Modes
log₁₀, ln, log₂, any custom base, antilog, and a reference table.
Instant Results
All logarithm values update immediately as you type.
Step-by-Step
Full working shown using the change-of-base formula with verification.
Mobile Optimized
Clean layout designed for comfortable use on any device.
✎ Real-World Example
Step 2: Difference = 6.3 − 4.0 = 2.3 units
Step 3: Intensity ratio = 10^2.3 = antilog(2.3)
Step 4: 10^2.3 ≈ 199.5
Q Frequently Asked Questions
Log Calculator - Solve Any Logarithm Easily, Free and Online
"What power do I raise this number to, to get that number?" That's genuinely the entire concept behind a logarithm, even though the notation — log base 3 of 81, for instance — tends to make it look far more intimidating than it actually is. A free log calculator online strips away that intimidation entirely, handling the computation instantly so you can focus on understanding what the answer actually means rather than getting stuck on the arithmetic.
A genuinely easy log calculator doesn't just default to base 10 either. Natural logs, common logs, and completely custom bases like log base 7 or log base 2 all come up regularly in coursework and professional work, and a tool worth using needs to handle every one of them without forcing you to memorize a separate formula for each case.
What a Logarithm Actually Represents
log_b(x) = y, which means b^y = x
This single relationship is the foundation of every logarithm that exists. Asking "log base b of x" is really just asking what exponent y makes b raised to that power equal x — nothing more complicated than that, despite how the symbol itself often gets treated as one of the scarier parts of algebra.
Common Log vs Natural Log
Two specific bases show up so often that they get their own shorthand notation, and knowing the difference matters more than most people realize.
log(x) = log_10(x) — when no base is written, it's almost always assumed to be base 10, which is why this is called the "common" log and appears constantly in chemistry, acoustics, and earthquake measurement scales.
ln(x) = log_e(x) — the natural logarithm uses Euler's number (e ≈ 2.71828) as its base, and it shows up throughout calculus, compound growth models, and any calculation involving continuous rather than stepped change.
Solving a Log With Any Base
Most calculators, including basic phone apps, only offer common and natural log buttons — leaving you stuck if a problem asks for log base 5 or log base 12. This free calculator gets around that limitation using the change of base formula:
log_b(x) = log(x) ÷ log(b)
This lets you calculate a logarithm in literally any base using only common or natural logs behind the scenes, which is exactly how this tool computes an unusual base instantly instead of requiring a specialized formula for every possible number.
How This Free Log Calculator Works
Enter the number you're taking the log of, then enter your base — 10 by default for a common log, or switch to e for a natural log with a single click. For any other base, just type it directly into the base field, and the change of base formula applies automatically behind the scenes. The result appears instantly, and expanding "Steps" shows exactly how that formula was applied, which is genuinely useful when a class or report requires showing the full method rather than just stating a final decimal.
Working Through a Chemistry Problem: Finding pH
pH is defined directly using a base-10 logarithm: pH = −log[H⁺], where [H⁺] represents hydrogen ion concentration in moles per liter. Take a solution with a hydrogen ion concentration of 0.0001 moles per liter.
Written in scientific notation, that's 1 × 10⁻⁴. Applying the formula gives pH = −log(1 × 10⁻⁴). Since log(10⁻⁴) equals exactly −4, the calculation becomes pH = −(−4), landing on a final pH of 4 — clearly acidic, since anything below 7 falls on that side of the scale. This exact calculation runs constantly in chemistry labs, water treatment plants, and even home pool maintenance kits, all built on the same logarithm this calculator solves in a fraction of a second.
A Second Example: Solving for Time
Suppose a bacteria population doubles every hour, starting at 100 and needing to reach 12,800. Setting up 100 × 2^t = 12,800 and dividing both sides by 100 gives 2^t = 128. Taking log base 2 of both sides isolates t, and t = log_2(128) comes out to exactly 7 hours — a problem that would take considerably longer to solve through trial and error than by directly applying a logarithm to pull that exponent out.
| x | log₁₀(x) | ln(x) | log₂(x) |
|---|---|---|---|
| 0.001 | -3 | -6.90775528 | -9.96578428 |
| 0.01 | -2 | -4.60517019 | -6.64385619 |
| 0.1 | -1 | -2.30258509 | -3.32192809 |
| 0.5 | -0.30103 | -0.69314718 | -1 |
| 1 | 0 | 0 | 0 |
| 2 | 0.30103 | 0.69314718 | 1 |
| 3 | 0.47712125 | 1.09861229 | 1.5849625 |
| 4 | 0.60205999 | 1.38629436 | 2 |
| 5 | 0.69897 | 1.60943791 | 2.32192809 |
| 8 | 0.90308999 | 2.07944154 | 3 |
| 10 | 1 | 2.30258509 | 3.32192809 |
| 16 | 1.20411998 | 2.77258872 | 4 |
| 20 | 1.30103 | 2.99573227 | 4.32192809 |
| 50 | 1.69897 | 3.91202301 | 5.64385619 |
| 100 | 2 | 4.60517019 | 6.64385619 |
| 256 | 2.40823997 | 5.54517744 | 8 |
| 1000 | 3 | 6.90775528 | 9.96578428 |
| 10000 | 4 | 9.21034037 | 13.28771238 |
Real Situations Where Logs Actually Come Up
- Chemistry: pH calculations and reaction rate equations
- Acoustics: decibel scales, which are logarithmic rather than linear by design
- Earthquake measurement: the Richter scale compresses an enormous energy range using logs
- Finance: figuring out how long an investment takes to reach a target value
- Computer science: analyzing algorithm efficiency, where logarithmic time complexity signals a highly efficient process
Logs and Exponents Are Mirror Images
Where an exponent answers "what do I get from raising this base to this power," a logarithm answers the exact reverse — "what power gets me to this result." Because of that tight relationship, this tool pairs naturally with the Exponent Calculator for working in the opposite direction, and problems mixing both operations together often benefit from the broader Scientific Calculator, which handles logs and exponents within a single continuous expression.
Log Calculator — Logarithm Rules
| Rule | Formula |
|---|---|
| Product rule | log(xy) = log(x) + log(y) |
| Quotient rule | log(x/y) = log(x) − log(y) |
| Power rule | log(xⁿ) = n × log(x) |
| Change of base | log₉(x) = log(x) ÷ log(b) |
| Log of 1 | log₉(1) = 0 |
| Log of base | log₉(b) = 1 |
Working Backward With an Antilog
Sometimes the question runs in reverse — you know a logarithm's result and need to find the original number it came from. That's exactly what the Antilog Calculator is built for. If log(x) = 3, the antilog undoes that operation and confirms x equals 1000, recovering the original value rather than computing something new.
Logs Trapped Inside Algebra Problems
Logarithms show up constantly in algebra whenever an unknown variable gets stuck inside an exponent, since taking a log of both sides is often the only clean way to pull that variable out algebraically. For equations that combine logs with other operations, the Equation Solver and Algebra Solver can finish the problem once the logarithm itself has been calculated correctly.
Formatting the Final Result
Log calculations often connect to numbers that are extremely large or extremely small, which is exactly the territory the Scientific Notation Calculator handles cleanly. And when a final result needs sensible rounding for a lab report, the Significant Figures Calculator helps determine how much precision that answer actually deserves.
A Mix-Up Worth Avoiding
Plenty of people assume "log" always defaults to base 10 in every context, but that's not universally true — many programming languages and higher-level math courses default to natural log whenever "log" is written without a specified base. Double-checking which convention a specific textbook or tool is using before assuming the base saves a lot of unnecessary confusion.
Frequently Asked Questions
Is this log calculator actually free, with no sign up?
Yes, it's completely free with no sign up, no account, and no download. It works instantly online, directly in your browser.
What's the easiest way to understand what a logarithm means?
Think of it as answering one question: what exponent do I need to raise this base to, in order to reach this number? Once that clicks, most logarithm problems become far less intimidating.
How do I calculate a log with an unusual base like log base 5?
Use the change of base formula: log_5(x) = log(x) ÷ log(5). This calculator applies that automatically the moment you enter a custom base.
Can a logarithm result be negative?
Yes, if the number you're taking the log of is between 0 and 1, the result comes out negative, representing a negative exponent needed to bring the base down below 1.
Why is log of a negative number undefined?
Because no real exponent applied to a positive base can ever produce a negative number or zero, logarithms of non-positive numbers simply don't exist within the real number system.
Explore more calculators: Scientific Calculator | Exponent Calculator | Antilog Calculator | Algebra Solver
Reviewed by Kromo Tools Team
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