Logarithm Calculator
A logarithm answers one question: what power do you raise the base to in order to reach x? Enter a value and a base and this works out log_b(x), shows the common, natural and binary logarithms of the same number so the change-of-base relationship is visible rather than asserted, and solves the inverse — given a base and a logarithm, it returns the x they came from.
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Logarithm of 1000 to base 10
3
Natural logarithm ln(x)
6.907755
Details
Updates as you typeThe number you want the logarithm of. Must be greater than zero.
Use 10 for common logs, 2 for binary logs, or 2.718281828 for natural logs. Must be positive, and cannot be 1.
The inverse question: if the logarithm were y, what was x? The answer is b raised to this power.
Summary
Logarithm
3
- Common logarithm log₁₀(x)
- 3
- Natural logarithm ln(x)
- 6.907755
- Binary logarithm log₂(x)
- 9.965784
- Change of base: ln(x) ÷ ln(b)
- 3.000000
- Inverse: x when the logarithm is y
- 1,000
- Check: b raised to the answer
- 1,000
- Is x an exact power of b?
- Yes — b^3 = 1000
- 1000 is an exact power of 10, so the logarithm is the whole number 3.
- Change of base alone returns 2.9999999999999996 for this one, because neither natural logarithm is exact in binary and the residue survives the division. The whole number above is confirmed the other way round: 10 raised to 3 gives 1000.
How this is calculated
- Value x
- 1,000.000000
- Base b
- 10.000000
- ln(x)
- 6.907755
- ln(b)
- 2.302585
- Change of base: log_b(x) = ln(x) ÷ ln(b)
- 6.907755279 ÷ 2.302585093 = 3
- Answer y = log_b(x)
- 3
- Check by exponentiating: b^y = x
- 10^3 = 1000
- Known logarithm y
- 3.000000
- Inverse: x = b^y
- 10^3 = 1000
Compare scenarios
See how one change moves the result
- CurrentYour inputs as they stand3Current
- Value (x)1,2503.096910
- Base (b)12.52.734955
Frequently asked questions
What is the change-of-base formula?
log_b(x) = ln(x) ÷ ln(b), and the same identity works with any logarithm on top and bottom — log10(x) ÷ log10(b) gives the identical answer. It exists because calculators and programming languages only ship a couple of fixed bases, so every other base is reached by dividing two logarithms you already have. The worked steps show the two natural logarithms and the division that produced the answer.
Why can you not take the logarithm of zero or a negative number?
A positive base raised to any real power is always positive: 10 to a large negative power gets very small but never reaches zero, and never crosses below it. So there is no real exponent that produces 0 or -5, and the logarithm of those values does not exist over the real numbers. This calculator reports an error against the x field rather than returning a meaningless figure.
Why is base 1 not allowed?
Because 1 raised to any power at all is still 1. There is no exponent that turns 1 into 8, so log base 1 of 8 has no answer — and log base 1 of 1 has too many, since every exponent works. The base must also be positive, because a negative base raised to a fractional power leaves the real numbers entirely.
Why do some calculators show log10(1000) as 2.9999999999999996?
Because they compute it as ln(1000) ÷ ln(10). Neither natural logarithm can be stored exactly in binary floating point, and the tiny error in each survives the division. This calculator uses the dedicated base-10, base-2 and natural routines where they apply, and for any other base it only rounds a near-whole answer to the whole number after confirming that raising the base back to it reproduces x exactly.
What is an antilogarithm?
It is the inverse operation: if log_b(x) = y, then the antilog is x = b raised to the power y. The two undo each other, which is why raising the base back to the answer returns the value you started with. The known-logarithm field on this page runs that direction, so you can go from a logarithm back to the number it describes.
What are the common, natural and binary logarithms?
They are the three bases that come up constantly. The common logarithm uses base 10 and counts digits, so log10 of a million is 6. The natural logarithm uses base e, roughly 2.718281828, and is the one that appears in growth and decay. The binary logarithm uses base 2 and counts doublings, so log2 of 1024 is 10. All three are shown for whatever value you enter.