About the Log Calculator
A logarithm answers the question "what exponent produces this number?" A log calculator evaluates that in any base — natural, base 10, base 2, or one you choose. Logarithms compress vast ranges into workable numbers, which is why sound, earthquakes, acidity, and algorithmic complexity are all measured logarithmically.
The formula
log_b(x) = ln(x) / ln(b)b is the base and x the value. The identity shown is the change-of-base formula, which is how a logarithm in any base is computed from natural logarithms.
How to use this calculator
- 1Enter your Value (x). The field starts at
100, which you can overwrite. - 2Enter your Base (e.g. 10, 2, e=2.718). The field starts at
10, which you can overwrite. - 3Read the result straight away — it recalculates as you type, so there is no button to press. Use Share to copy a link that reopens the page with your exact numbers filled in.
Worked example
| Input | Value |
|---|---|
| Value (x) | 100 |
| Base (e.g. 10, 2, e=2.718) | 10 |
Result
log10(100) = 2.000000
ln(100) = 4.605170
log₁₀(100) = 2.000000
log₂(100) = 6.643856
Those are the values the page loads with, so you can reproduce this result yourself and then change one field at a time to see what drives the outcome.
Understanding your result
A logarithm is the inverse of exponentiation. Because 10³ = 1000, log₁₀ 1000 = 3. This inverse relationship is what makes logs useful for solving equations where the unknown sits in the exponent — compound interest problems asking how long money takes to double, for instance.
The compressive property is the practical one. Adding 1 to a base-10 logarithm means multiplying the original number by ten, so a scale that runs 1 to 10 in log terms covers 10 to 10 billion in real terms. That is why a magnitude 7 earthquake releases roughly 32 times the energy of a magnitude 6, and why decibels can span the entire range of human hearing in about 120 steps.
Things worth knowing
- Logarithms of zero and negative numbers are undefined in the real numbers. log 1 = 0 in every base.
- log(a · b) = log a + log b, which is the property that made slide rules and log tables work.
- ln uses base e ≈ 2.71828 and appears in continuous growth, decay, and calculus.
- The change-of-base formula lets any calculator handle any base: log₂ 8 = ln 8 / ln 2 = 3.
- In computer science, log usually means base 2, whereas in engineering it usually means base 10. Check the convention.
Frequently asked questions
What is a logarithm in plain terms?+
The exponent needed to raise a base to a given number. log₁₀ 100 = 2 because 10² = 100. It is the inverse operation of raising to a power.
What is the difference between log and ln?+
ln is the natural logarithm to base e ≈ 2.718, which arises in continuous growth and calculus. log normally means base 10, used in scientific notation and measurement scales.
Why is the logarithm of zero undefined?+
Because no exponent applied to a positive base produces zero. As the input approaches zero the logarithm falls without limit, so there is no finite value to assign.
How do I calculate a logarithm in an unusual base?+
Use the change-of-base formula: divide the natural log of your value by the natural log of the base. log₇ 343 = ln 343 / ln 7 = 3.