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Log Calculator

Logarithms in Base 10, e, 2 or Any Base, Plus Antilog

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How the Log Calculator works

This log calculator finds the logarithm of any positive number in base 10, base e, base 2 or a base you choose, and lists log₁₀, ln and log₂ together so you can compare them. Switch to antilog to go the other way and raise the base to a power. Each answer comes with a check that undoes it.

A logarithm answers the question "what power gives this number?" Since 10³ = 1000, log₁₀(1000) = 3. Logs turn multiplication into addition, which is why they measure things that span huge ranges: decibels, the Richter scale, pH and star brightness all use them. Base 10 suits everyday scales, base e is the natural log of growth and calculus, and base 2 counts bits and doublings in computing.

How the log calculator works

The formula is log_b(x) = ln(x) / ln(b), the change of base rule, so any base comes down to natural logs. The calculator rounds a result to a whole number when it is within a trillionth of one, so log₂(64) shows 6 rather than 5.999999. The antilog is b^x, and the check line raises the base to the answer to confirm it.

First example. log₁₀(1000) = 3, because 10 × 10 × 10 = 1000. Its ln is 6.907755 and its log₂ is 9.965784.

Second example. log₂(64) = 6, because 2⁶ = 64. In computing, that means 64 values fit in 6 bits.

How many times must 1 double to pass a million? log₂(1,000,000) ≈ 19.93, so 20 doublings.

Logarithms of common numbers in base 10, e and 2

x log₁₀ x ln x log₂ x
0.001 -3 -6.9078 -9.9658
0.01 -2 -4.6052 -6.6439
0.1 -1 -2.3026 -3.3219
0.5 -0.301 -0.6931 -1
1 0 0 0
2 0.301 0.6931 1
3 0.4771 1.0986 1.585
4 0.6021 1.3863 2
5 0.699 1.6094 2.3219
6 0.7782 1.7918 2.585
7 0.8451 1.9459 2.8074
8 0.9031 2.0794 3
9 0.9542 2.1972 3.1699
10 1 2.3026 3.3219
16 1.2041 2.7726 4
20 1.301 2.9957 4.3219
32 1.5051 3.4657 5
50 1.699 3.912 5.6439
64 1.8062 4.1589 6
100 2 4.6052 6.6439
256 2.4082 5.5452 8
1,000 3 6.9078 9.9658
1,024 3.0103 6.9315 10
1,000,000 6 13.8155 19.9316

Tips and pitfalls

FAQ

What is a logarithm?

A logarithm is the exponent you need to raise a base to in order to get a number. log₁₀(100) = 2 because 10² = 100, and log₂(8) = 3 because 2³ = 8. Logs and exponents are inverses, so each undoes the other in the same base.

What is the difference between log and ln?

log usually means base 10 and ln means the natural log, base e ≈ 2.71828. They differ by a constant factor: ln(x) ≈ 2.302585 × log₁₀(x). The natural log appears in growth, decay and calculus, while base 10 suits scales such as decibels and pH.

How do you calculate a log with a different base?

Use the change of base rule: log_b(x) = log(x) / log(b), with any common base on top and bottom. So log₃(81) = log(81) / log(3) = 1.908485 / 0.477121 = 4. Choose Custom base and type the base to skip the division.

What is an antilog?

The antilog reverses a logarithm by raising the base to the given power. The antilog of 2 in base 10 is 10² = 100, and the antilog of 1 in base e is e ≈ 2.718282. Pick Antilog in the Find box and enter the log value as the number.

Why is log of 1 equal to zero?

Any base raised to the power 0 is 1, so the log of 1 is 0 in every base. Numbers between 0 and 1 have negative logs, such as log₁₀(0.01) = -2, and the log heads toward minus infinity as the number approaches zero.

See also: all Arithmetic tools, Number Theory, Algebra and Geometry, Statistics, Trigonometry and Calculus.

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