Logarithm calculator
Compute the logarithm in any base and read ln, log₁₀ and log₂ at the same time.
Calculator inputs
Results
Enter your values and press “Calculate” to see the result.
In short
- What it calculates
- Compute the logarithm in any base and read ln, log₁₀ and log₂ at the same time.
- Formula used
log_b(a) = c ⟺ b^c = a- Example
- log₁₀(1000)
Logarithm calculator
A logarithm answers the question “what exponent must the base be raised to in order to get this number?”.
The calculator returns the logarithm in the base you choose plus the three most used ones: natural, common and binary.
How it works
Logarithms exist only for positive numbers, and the base must be positive and different from one. Unusual bases are handled with the change of base formula using natural logarithms.
Formula
log_b(a) = c ⟺ b^c = a
Change of base: log_b(a) = ln(a) / ln(b)
log(x·y) = log(x) + log(y)
log(xⁿ) = n · log(x)
Worked example
log₁₀(1000)
10 raised to what gives 1000?
10³ = 1000, so the answer is 3.
ln(1000) = 6.907755 and log₂(1000) = 9.965784
Explanation
A logarithm answers the inverse question of a power
Where a power asks what two to the tenth equals, the logarithm asks what exponent two must be raised to in order to reach 1,024. They are the same relationship read backwards, which is why anything one does the other undoes.
Why they turn multiplication into addition
The logarithm of a product is the sum of the logarithms. That property is what let Napier invent them in the seventeenth century to simplify astronomical calculations, and it underpins the slide rules used until calculators arrived.
Logarithmic scales in the real world
The Richter scale, decibels, pH and stellar magnitude are all logarithmic. That is why a magnitude seven earthquake releases about thirty-two times more energy than a magnitude six, not slightly more. When something spans many orders of magnitude, a logarithmic scale is the only readable one.
Natural and common logarithms
The natural logarithm uses the number e as its base, and appears spontaneously in continuous growth, compound interest and radioactive decay. The common logarithm uses base ten and is convenient for reading orders of magnitude.
Frequently asked questions
Why is there no logarithm of a negative number?
Because no real power of a positive base ever produces a negative result.
What is the difference between ln and log?
ln uses base e (2.71828…) while log usually means base 10. In computing, log₂ is the most common.
How much more energy does a magnitude 7 earthquake release than a 6?
About thirty-two times more. The scale is logarithmic, so each point multiplies the energy released rather than adding to it.
When does the natural logarithm appear?
Whenever there is continuous growth or decay: compound interest capitalised without interruption, radioactive decay or population models.
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