Logarithm Calculator

Written byOleh Soloninupdated 5 October 2026

Value

Base

10 for the common log, 2 for binary; the natural ln is always shown.

Result

Log to the base3
Check10^3 = 1000
Common log (log₁₀)3
Natural log (ln)6.907755
Binary log (log₂)9.965784

The power you raise the base to

The logarithm of a number to any base, shown alongside the common log, natural log and binary log of the same number, with a check by raising the base to the power.

The logarithm of b to base a is the power you raise a to in order to get b: logₐ b = c when aᶜ = b. So log₁₀ 1000 = 3 because 10³ = 1000, and log₂ 8 = 3 because 2³ = 8.

The common logarithm has base 10 and the natural logarithm, ln, has base e ≈ 2.71828. The calculator shows both, plus log₂, next to the logarithm in the base you chose.

How to use

Enter the number; it must be greater than zero.
Enter the base: 10 for the common logarithm, 2 for binary, or any positive base other than 1.
Read the logarithm, the check, and log₁₀, ln and log₂ of the number.

Log rules and changing the base

logₐ(xy) = logₐ x + logₐ y; logₐ(x/y) = logₐ x − logₐ y; logₐ(xⁿ) = n · logₐ x. The log of 1 is zero in any base, and the log of the base itself is one.

Change of base: logₐ b = ln b ÷ ln a = log₁₀ b ÷ log₁₀ a. That is how the calculator works it out: log₃ 20 = ln 20 ÷ ln 3 ≈ 2.9957 ÷ 1.0986 ≈ 2.7268.

Worked out with this calculator, rounded to six decimal places.
Number	log₁₀	ln	log₂
1	0	0	0
2	0.30103	0.693147	1
10	1	2.302585	3.321928
100	2	4.60517	6.643856
1000	3	6.907755	9.965784
1024	3.0103	6.931472	10

What this calculator shows

The logarithm of the number to your base
A check: the base raised to that power
The common log, natural log and binary log of the same number

What to keep in mind

Logarithms are defined for positive numbers only.
The base must be positive and not equal to one.

FAQs

What is the difference between log and ln?

log usually means base 10 and ln base e ≈ 2.71828. They differ by a constant factor: ln x ≈ 2.302585 × log₁₀ x.

How do I take a logarithm to base 2?

Enter base 2, or read the log₂ line, which is always shown. log₂ 1024 = 10 because 2¹⁰ = 1024.

How do I change the base?

Divide logarithms in any common base: logₐ b = ln b ÷ ln a. For example, log₃ 20 = ln 20 ÷ ln 3 ≈ 2.7268.

What is the log of 1?

Zero in every base, because any allowed base to the power zero is one.

How is pH a logarithm?

In the school definition pH is minus the common logarithm of the hydrogen ion concentration, pH = −log₁₀[H⁺]; strictly it is their activity, which in dilute solutions is close to the concentration. At 10⁻⁷ mol/l the pH is 7.

What is the logarithm of 1000 to base 10?

Three, because ten cubed is a thousand. The calculator also gives the natural logarithm of the same number, 6.9078, and the binary one, 9.9658.

Why three different bases?

Base ten reads orders of magnitude and decibels, the natural log appears wherever growth is continuous, and base two answers how many doublings something takes: 9.97 says a thousand is nearly ten doublings.

Worked example

The common log of a thousand

Input: Number 1000, base 10

Output: 3

Note: Check: 10³ = 1000. The natural log of the same number is 6.907755 and the binary log 9.965784.

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Logarithm Calculator

Result

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