Logarithm Calculator
Calculate logarithms in any base with presets for log10, natural log and log2 plus antilog and change of base formula
Logarithm
2.000000
log10.00(100) = 2.000000
Antilogarithm (Inverse)
100.000000
10.00^2.000000
Common Log Values
log10(100) = 2.000000
ln(100) = 4.605170
log2(100) = 6.643856
Change of Base Formula
log_b(x) = ln(x) / ln(b) = log10(x) / log10(b)
log_10.00(100) = ln(100) / ln(10.00)
What a Logarithm Actually Is
A logarithm answers the question: 'what power do I raise this base to in order to get this number?' log₁₀(1000) = 3 because 10³ = 1000. log₂(8) = 3 because 2³ = 8. ln(x) is shorthand for log base e, where e ≈ 2.71828. The natural log shows up in growth and decay problems because e is the base of continuous compounding.
If your calculator only has log10 and ln but you need a different base, use the change of base formula: log_b(x) = ln(x) / ln(b), or equivalently log(x) / log(b). So log₅(625) = log(625) / log(5) = 2.7959 / 0.6990 = 4. The calculator on this page handles any base directly so you do not have to do the conversion by hand.
When You Meet Logs in Real Problems
A-level students hit logs first when solving exponential equations like 2^x = 50. Take the log of both sides, drop the exponent down using the power rule, divide. log(2^x) = log(50), so x × log(2) = log(50), so x = log(50) / log(2) ≈ 5.644. That is the calculation behind every compound interest doubling-time, every radioactive decay half-life, every population growth model.
Logs are also the basis of the decibel scale (sound), the Richter scale (earthquakes), and pH (chemistry). All three compress huge ranges into manageable numbers. A 6.0 earthquake releases 31.6 times more energy than a 5.0, because the scale is logarithmic in base 10. If you ever need very large or very small numbers in compact form, [Scientific Notation Converter](/scientific-notation-converter) handles the related job of expressing them as a × 10^n.
Common Logarithm Values
| Expression | Meaning | Value |
|---|---|---|
| log₁₀(100) | 10^? = 100 | 2 |
| log₁₀(1000) | 10^? = 1000 | 3 |
| log₂(64) | 2^? = 64 | 6 |
| ln(e) | e^? = e | 1 |
| ln(1) | e^? = 1 | 0 |
| log₅(625) | 5^? = 625 | 4 |
Frequently Asked Questions
What is the difference between log and ln?
On most calculators, 'log' means log base 10 and 'ln' means log base e (natural log). On some scientific calculators and in computer science contexts, 'log' may default to log base 2. The calculator here lets you pick the base explicitly so there is no ambiguity.
Why can I not take the log of zero or a negative number?
There is no power you can raise a positive base to that produces zero or a negative result. 10^x is always positive for any real x. So log(0) and log(-5) are undefined in the real numbers. (In complex analysis, you can extend the definition, but that is well past the level of a standard calculator.)
What is the change of base formula?
log_b(x) = log_a(x) / log_a(b), where a is any base your calculator already knows. Most use a = 10 or a = e. So log₇(50) = ln(50) / ln(7) ≈ 3.912 / 1.946 ≈ 2.011. Useful when your calculator is hardcoded to log10 and ln only.
What is an antilog?
The inverse of a logarithm. If log_b(x) = y, then antilog_b(y) = x. So if log₁₀(x) = 2, then x = 10² = 100. The antilog button on a calculator is essentially 10^x or e^x depending on which log you used.