Log Calculator
Enter a number and a base — 10 by default — and get its logarithm, with the natural log, base-2 and base-10 values for the same number and a check that raising the base to the answer brings your number back.
Change of base: log base b of x is ln x ÷ ln b, which is how a logarithm in any base is worked out by hand.
Formula and calculation
Worked example
The log of 1,000 in base 10 is 3, because 10³ = 1,000. The same number as a natural log is 6.9078 and as a base-2 log it is 9.9658, and raising 10 back to the answer gives 1,000 — the check.
Logarithms you can do in your head
Powers of ten are the easy ones: log10(10) = 1, log10(1,000) = 3, log10(1,000,000) = 6, and log10(0.001) = −3 because 0.001 is 1 ÷ 1,000. Anything between 10 and 100 has a log between 1 and 2; the leading digit sets the whole part. Two anchors worth remembering are log10(2) ≈ 0.30103 (which is why doubling ten times is almost exactly a factor of 1,000) and ln(2) ≈ 0.693, the constant behind half-lives and compound interest.
Logs undo powers
A logarithm answers the question a power asks backwards: given 10³ = 1,000, the log tells you that the exponent was 3. The same idea runs through roots — the square root of 1,000,000 is 1,000, and the square root calculator writes that as a fractional exponent of 1/2 — so powers, roots and logs are three ways of asking one question about a base and an exponent.
What the results mean
- Result
- The logarithm of your number in the base you chose: the power that the base must be raised to in order to give that number. log10(1000) = 3 because 10³ = 1000.
- Natural log (ln x)
- The same number as a natural logarithm, ln x, which uses the constant e = 2.718281828… as its base. It is the log that turns up in growth, decay and calculus.
- Log base 2
- The same number in base 2 — how many doublings or halvings it takes to get there. log2(1024) = 10, which is why 1,024 is such a familiar number in computing.
- Log base 10
- The same number in base 10, the common log used for decibels, pH and the Richter scale. For whole powers of ten it is just the number of zeros: log10(1000) = 3.
- Base raised to the result
- The base raised back to the answer — a check that the logarithm is right. It should come back to the number you entered, give or take the last floating-point digit.
- Calculation
- The logarithm written out as the power it stands for, so you can check it by hand: log10(1000) = 3 because 10^3 = 1000.
Frequently asked questions
What is log base 10 of 1000?
3, because 10³ = 1000. The common log counts the multiplications by ten, so log10(100) = 2, log10(1000) = 3 and log10(0.001) = −3. Enter 1000 with base 10 and the calculator shows both the answer and the check, 10^3 = 1000.
Why is the log of 1 always 0?
Because every base raised to the power 0 is 1, so log_b(1) = 0 for any allowed base. On this page, entering 1 gives 0 for the result and 0 for the natural log and the base-2 and base-10 rows as well.
What is a natural log (ln)?
A logarithm in base e, the constant 2.718281828… that describes continuous growth. ln(e) = 1, ln(1) = 0 and ln(10) = 2.302585. It is the same logarithm as any other, just in a different base, which is why the ln row sits under the main result.
Can I take the logarithm of a negative number or of zero?
Not as a real number. A positive base raised to any power stays positive, so no exponent can produce 0 or a negative result. This calculator reports the problem instead of returning a meaningless number; the complex-valued answers are outside what a calculator of this kind can show.
What is the difference between ln, log and log2?
Only the base. ln uses e, log2 uses 2 and log10 uses 10. They are all proportional, which is the change-of-base rule: ln x ÷ ln b converts between them. The natural log is 2.302585 times the common log of the same number, since ln 10 = 2.302585.