Log Calculator (Logarithm)
Evaluate a logarithm in base 10, base 2, base e, or any base you choose (with the change-of-base step shown).
Details
Must be greater than zero.
log₁₀(1,000) =
3
The power the base must be raised to in order to reach 1,000
Natural log (ln x)
6.907755
Common log (log₁₀ x)
3
Binary log (log₂ x)
9.965784
Base you chose
10
This works out a log in any base. Choose base 10, e (natural log) or 2, or type your own, then enter the number.
It returns the answer along with the same number's natural log, common log and binary log, and shows the working so you can see whether it came out as a clean whole power or needed the change-of-base rule.
What is a log?
A log (short for logarithm) answers one question: how many of this number do I multiply together to get that number?
Say you multiply three 2s: 2 × 2 × 2 = 8. You used three 2s, so the log of 8, using 2s, is 3.
That is the whole idea. Powers ask 'what do I get?'. Logs ask 'how many did I use?'. They are the same fact read in opposite directions, the way subtraction reverses addition, so a log undoes what the exponent calculator does.
The small number tucked under the word log is the base. The number in brackets is where you want to end up. Read it as: how many 2s multiply together to make 8? Three of them.
What to enter
- Base (b)
- The number you keep multiplying. Choose 10, e, or 2 from the list, or pick Custom base for anything else.
- Custom base value
- Only shows if you chose Custom. Any positive number except 1. One is banned because multiplying 1s only ever gives you 1, so you could never reach anything else.
- Number (x)
- The number you want to reach. It has to be bigger than zero.
The three logs that have their own names
- log — base 10
- Just written 'log' with no small number. Multiplying 10s. Used in science and engineering, and behind the decibel and pH scales.
- ln — base e
- Called the natural log. The base is e, about 2.718. Written 'ln' rather than logₑ. Turns up wherever something grows or shrinks smoothly, like compound interest.
- log₂ — base 2
- The computing one, because computers work in twos. It tells you how many times you can halve something.
What this assumes
The base must be positive and cannot be 1.
The number must be bigger than zero. Logs of 0 or of negative numbers do not exist in ordinary maths.
How to calculate a log
Turn the log back into a multiplication and the answer usually appears on its own.
- b
- The base: the number you keep multiplying
- x
- The number you want to reach
- y
- The answer: how many you needed
Say it out loud. log₁₀(100) is asking: how many 10s multiply together to make 100? 10 × 10 = 100, so the answer is 2.
Learn two freebies. The log of 1 is always 0, in any base, because you needed none of them. And the log of the base itself is always 1, because you needed just one.
Use the four rules. Multiplying inside a log becomes adding outside it, and dividing becomes subtracting. That is the trick logs were invented for. The rules are in the FAQs below.
Change the base if your calculator lacks it. Most calculators only have log and ln. For any other base, divide one by the other: log₂(32) = log(32) ÷ log(2). The scientific calculator has both keys.
See a worked example: how many 2s make 32?
- Base
- 2
- Number
- 32
Just count as you multiply: 2, then 4, then 8, then 16, then 32.
2 × 2 = 4 (that is two 2s). × 2 = 8 (three). × 2 = 16 (four). × 2 = 32 (five).
You used five 2s, so the answer is 5.
Checking it with the change of base rule gives the same thing: log(32) ÷ log(2) = 1.5051 ÷ 0.3010 = 5.
log₂(32) = 5
Frequently asked questions
Multiply becomes add: log(x × y) = log(x) + log(y). Example: log(2) + log(5) = 0.3010 + 0.6990 = 1, and 2 × 5 = 10, whose log is 1.
Divide becomes subtract: log(x ÷ y) = log(x) − log(y). Example: log(10 ÷ 2) = 1 − 0.3010 = 0.699.
A power comes out to the front: log(xⁿ) = n × log(x). Example: log(2⁶) = 6 × 0.3010 = 1.806.
Change of base: log_b(x) = log(x) ÷ log(b). This is how you get any base out of a calculator that only has log and ln.
Only the base. Plain log means you are multiplying 10s. ln means you are multiplying e's, where e is about 2.718.
Use log for things measured in powers of ten, like decibels or pH. Use ln for things that grow or shrink smoothly over time.
Divide one log by another: log₂(x) = log(x) ÷ log(2). It works with ln too.
For example log₂(32) = 1.5051 ÷ 0.3010 = 5. The same trick works for any base.
Because no amount of multiplying a positive number ever gets you there. Multiply 10s and you get 10, 100, 1000. Divide instead and you get 0.1, 0.01, 0.001, creeping towards zero but never landing on it and never going below.
So there is no answer to give, and the calculator says the input is out of range rather than inventing one.
For squashing huge ranges of numbers into something readable. Earthquake magnitude, decibels and pH are all logs, which is why a magnitude 6 quake releases roughly 32 times the energy of a magnitude 5 rather than 20% more.
In computing, log₂ tells you how many times a list can be halved, which is why finding a name in a sorted list of a million takes about 20 checks instead of a million.
Problems people actually run into
Reading a log scale as if it were a normal one
This is the mistake that matters outside the classroom. On a log scale each step multiplies rather than adds.
A 60 dB sound is not a bit louder than 50 dB, it carries ten times the intensity. pH 5 is ten times more acidic than pH 6. And a chart of cases or prices drawn on a log axis turns explosive growth into a gentle slope, which is exactly why it is used and exactly how it misleads anyone who did not check the axis.
Results are estimates for general information only and are not professional financial, medical, or legal advice. Read our full disclaimer.
Last updated: August 29, 2026