Use this log calculator to solve common log, natural log, log base 2, and custom-base logarithms.
Last updated
Logarithm calculator for log base 10, ln, log base 2, and full log-equation checks This log calculator now solves the full `log_b(x)=y` relationship: find the logarithm, recover the original value, or solve the base directly when the other two parts are known.
Solve mode
Base presets
Quick examples
Result
2
log_10(100) = 2, meaning 10 must be raised to 2 to produce 100.
Expression
log_10(100) = 2
Inverse check
10^2 = 100
Scientific notation
1.000000e+2
Common log base
Base 10 is the standard common logarithm used in scientific notation, pH, and decibel-style work.
Common log
2.000000
Natural log
4.605170
log₂ equivalent
6.643856
Same argument across common bases
Useful when you want to compare log base 10, ln, log base 2, and a custom base for the same argument.
Base
Expression
Result
Base 10
log_10(100)
2.000000
Base e
log_e(100)
4.605170
Base 2
log_2(100)
6.643856
Selected-base reference sheet
These checkpoint rows make it easier to sanity-check whether your result looks like a clean power, a reciprocal, or a fractional-power case.
Reference
Expression
Argument
Log result
y = -3
log_10(0.001) = -3
1.0000e-3
-3
y = -2
log_10(0.01) = -2
1.0000e-2
-2
y = -1
log_10(0.1) = -1
1.0000e-1
-1
y = 0
log_10(1) = 0
1.0000e+0
0
y = 1
log_10(10) = 1
1.0000e+1
1
y = 2
log_10(100) = 2
1.0000e+2
2
y = 3
log_10(1000) = 3
1.0000e+3
3
How to use this result
A logarithm answers the exponent question. If `log_b(x)=y`, then the same relationship can always be checked by reversing it to `b^y=x`. This page keeps both directions visible so you can compare base 10, natural log, binary log, and custom-base behaviour without switching tools.
Log calculator guide: solve log base 10, ln, log base 2, custom-base logs
A strong log calculator does more than return one number. It should handle common log, natural log, binary log, and custom-base logarithms, then help you work both directions of the equation `log_b(x)=y`. This page is built for that broader intent: log calculator, logarithm calculator, natural log calculator, log base 2 calculator, and log base n calculator.
What a logarithm means
A logarithm answers the exponent question. If `log_b(x)=y`, then `b^y=x`. In plain language, the logarithm tells you what power the base must be raised to in order to produce the argument.
That is why a logarithm is the inverse of an exponent. Exponentiation moves from base and power to a final value. A logarithm moves from base and final value back to the power. A good log calculator should make that inverse relationship obvious instead of leaving the user with one isolated output.
The core log formulas
Most log questions reduce to the same definition. Common log, natural log, binary log, and custom-base logs all follow it. The only thing that changes is the base.
The change-of-base formula matters because it lets a calculator move between bases cleanly. It also explains why many devices can still calculate a custom-base logarithm even if they only expose `ln` and `log` keys.
log_b(x) = y if and only if b^y = x
This is the defining relationship for any logarithm.
ln(x) = log_e(x)
The natural logarithm uses Euler's number e as its base.
log_b(x) = ln(x) / ln(b)
The change-of-base formula lets you evaluate a log in any valid base.
x = b^y
This inverse step is how you recover the original value from a logarithm result.
Common log, natural log, and log base 2
Common log usually means base 10. It is the form used most often for scientific notation, decibel-style scales, pH-style reasoning, and many classroom calculator questions. A common-log result of 2 means the argument is 100 because `10^2=100`.
Natural log, written `ln`, uses base e. This version appears constantly in calculus, continuous growth and decay, probability, and finance. Binary log, written `log₂`, uses base 2 and shows up in computer science, doubling and halving patterns, and algorithm complexity. These bases solve the same mathematical problem, but they fit different practical contexts.
Why a log calculator should solve more than one direction
Many searchers do not just want to know the value of a logarithm. They also want to recover the original number or solve for the base when the other two parts of the equation are known. Competitor pages often handle only the first of those jobs, which leaves users bouncing to an antilog calculator or doing the algebra separately.
This page keeps the full equation together. You can solve the logarithm itself, reverse it to find the original value, and solve for the base when the argument and log result are already known. That makes the page more useful for algebra checks, calculator-key verification, and custom-base problem solving.
Find the logarithm when the base and argument are known.
Find the original value `x` when the base and log result are known.
Find the base `b` when the argument and log result are known.
Compare the same argument or the same log result across common bases.
How to solve `log_b(x)=y` in each direction
If you are solving for the logarithm, you are asking for `y`. If you are solving for the original value, you reverse the log and compute `x=b^y`. If you are solving for the base, you rearrange the equation to `b=x^(1/y)` provided the setup is valid.
That last case is especially useful because many pages ignore it even though it is a natural part of logarithm algebra. If `log_b(81)=4`, then the base must be 3 because `3^4=81`. This is one of the cleanest ways to check whether you really understand the inverse relationship.
Domain rules people miss
Real-number logarithms have strict input rules. The argument must be positive. The base must also be positive and cannot equal 1. If either condition is broken, the result is not a valid real logarithm.
There are also subtler edge cases when solving for the base. If the log result is 0, then the equation demands `x=1` for every valid base, which means the base is not uniquely determined. If the argument is 1 and the log result is not 0, the setup is impossible. A good logarithm calculator should show a real warning instead of returning a fake answer.
What changes when the base is between 0 and 1
A base does not have to be greater than 1. Bases between 0 and 1 are still valid, but they reverse the intuition many users bring from base 10. With a fractional base, the log function decreases instead of increasing, and the sign of the result can look surprising if you are expecting the usual base-10 pattern.
This matters in custom-base work. It is one of the easiest places to make a sign mistake, especially when moving between the logarithm and the inverse `b^y=x`. Competitor pages often mention the valid-base rule but stop there. A better page explains how the behaviour actually changes.
Worked examples
For a common-log example, `log₁₀(100)=2` because `10^2=100`. For a natural-log example, `ln(e)=1` because `e^1=e`. For a binary-log example, `log₂(64)=6` because `2^6=64`.
The full-equation view adds two more useful examples. If `log₁₀(x)=2`, then `x=10^2=100`. If `log_b(81)=4`, then `b=81^(1/4)=3`. Those examples show why it is useful to keep the full log equation on one page instead of splitting the work across separate tools.
Why logarithms matter outside school math
Logarithms appear whenever multiplicative change is easier to reason about on a compressed scale. They are used in algorithm analysis, growth modeling, scientific notation, decibel-style comparisons, and many branches of physics, engineering, statistics, and finance.
That is why a log calculator is not just a classroom helper. It is a fast way to check exponent questions, verify inverse-log work, and compare how the same number behaves under common log, natural log, binary log, and a custom base.
The most common error is using the wrong base. A natural-log result must be reversed with base e, not base 10. Another common mistake is forgetting that `log₂`, `ln`, and `log₁₀` are not interchangeable even though they are all logarithms.
Users also often forget the domain rules, assume every logarithm should be positive, or expect the same intuition to hold for a base between 0 and 1. These are exactly the cases where comparison rows and inverse checks do more than a bare numeric output.
Frequently asked questions
What is a log calculator?
A log calculator evaluates logarithms such as common log, natural log, binary log, or a custom-base logarithm. A stronger one can also reverse the equation to find the original value or solve for the base when the other parts are known.
What is the difference between log and ln?
ln means natural log, which uses base e. In many contexts, plain log means base 10, though notation can vary by field. That is why checking the base matters before you interpret or reverse a result.
How do I calculate log base 2?
Use `log₂(x)` directly if your calculator supports it, or use the change-of-base formula `log₂(x)=ln(x)/ln(2)`. For example, `log₂(64)=6` because `2^6=64`.
How do I solve `log_b(x)=y` for x?
Reverse the logarithm by exponentiating the base: `x=b^y`. If `log₁₀(x)=2`, then `x=10^2=100`.
How do I solve `log_b(x)=y` for b?
Rearrange the equation to `b=x^(1/y)` when the setup is valid. For example, if `log_b(81)=4`, then `b=81^(1/4)=3`.
When is a logarithm undefined?
In real numbers, a logarithm is undefined if the argument is 0 or negative. A base is also invalid if it is 0, negative, or equal to 1.
Can a logarithm be negative?
Yes. For bases greater than 1, arguments between 0 and 1 produce negative logarithms. For example, `log₁₀(0.01)=-2` because `10^-2=0.01`.
Why does the base matter so much?
Because the base defines the exponential relationship you are undoing. The same argument can have very different log values in base 10, base e, base 2, or a custom base.
Can the base be less than 1?
Yes. Any positive base other than 1 is valid, including numbers between 0 and 1. But the function behaves differently there, so signs and monotonicity can feel less intuitive than they do with base 10.
What is the change-of-base formula used for?
It lets you evaluate a logarithm in any valid base using another log function you already have. The standard version is `log_b(x)=ln(x)/ln(b)`.
Is a logarithm the same as an antilog?
No. A logarithm finds the exponent, while the antilog reverses the process and recovers the original value. They are inverse operations.
Why do some calculators show both ln and log?
Because base e and base 10 are both common enough to deserve dedicated keys. `ln` is used heavily in higher mathematics and growth models, while `log` is often used for base-10 work.
