How do you calculate a log without a calculator?

For a base's exact powers, count: log₁₀(1000) is 3 because 10 × 10 × 10 = 1000. For anything else, bracket it — log₁₀(50) is between log₁₀(10) = 1 and log₁₀(100) = 2, closer to 2 — then use known values: log₁₀(50) = log₁₀(100) − log₁₀(2) = 2 − 0.30103 = 1.69897. With a calculator that has only ln or log keys, divide: log_b(x) = ln(x) ÷ ln(b).

What is the difference between log and ln?

ln is the natural logarithm, base e (about 2.71828); log written without a base usually means base 10 in school and engineering, though in some maths and computing texts a bare log means base e or base 2. They differ by a constant factor: ln(x) = 2.302585 × log₁₀(x). The page shows both for every input so there is no guessing.

How do I find a log in base 3, base 5 or any other base?

Choose 'Any other base', type the base, and the page divides ln(number) by ln(base). log₃(81) = ln(81) ÷ ln(3) = 4.394449 ÷ 1.098612 = 4, because 3⁴ = 81.

What is an antilog, and how do I do the reverse?

The antilog undoes the logarithm: if log₁₀(x) = 2.5 then x = 10^2.5 = 316.23. The check row on this page does exactly that for your answer, and the exponent calculator on this site raises any base to any power.

Why does log₂ matter?

Base 2 counts doublings and bits: log₂(1024) = 10 says 1024 is ten doublings of 1, and a number needs about log₂(n) bits to store. It appears in information theory, algorithm analysis and music (octaves are doublings of frequency).

Log Calculator (Any Base, ln, log₁₀, log₂)

Logarithm in base 10, e, 2 or any base you type, with the change-of-base working.

Inputs

Value (x)

01,000,000,000,000,000

Log base

Custom base

01,000,000

Used only when the base is set to 'Any other base'. Must be positive and not 1.

Include CalculateEveryday mark

log₅(36)

2.226566

Meaning: 5 raised to the power 2.226566 gives 36
Working (change of base): ln(36) ÷ ln(5) = 3.583519 ÷ 1.609438 = 2.226566
Log base 10: 1.556303
Natural log (ln, base e): 3.583519
Log base 2: 5.169925
Check: 5^2.226566 = 36

Use this calculator on your own site

Free embed

Two lines of HTML, unlimited views
Shows a small CalculateEveryday bar, with a credit link under the calculator

What each Log input means

Value (x): The number you want the logarithm of. It must be above 0; numbers between 0 and 1 have negative logarithms.
Log base: 10 for the common log (the log key on most calculators), e for the natural log (ln), 2 for the binary log, or any other base typed in the next box.
Custom base: Used when the base is 'Any other base': any positive number except 1, decimals included, such as 3, 5 or 0.5.

Log formula

log_b(x) = ln(x) ÷ ln(b) · meaning: log_b(x) = y exactly when b^y = x

\log_b x = \frac{\ln x}{\ln b} \qquad \log_b x = y \iff b^{y} = x

A logarithm answers one question: to what power must the base be raised to get this number? log₁₀(1000) is 3 because 10³ = 1000, and log₂(8) is 3 because 2³ = 8. Calculators only have two logarithm keys, ln (base e) and log (base 10), so any other base goes through the change-of-base formula: divide the natural log of the number by the natural log of the base. The page shows that division, the answer in base 10, base e and base 2 side by side, and a check that raises the base to the result and gets the number back. Whole-number answers are shown exactly rather than as 2.9999999, and a base between 0 and 1 is allowed — it just makes the logarithm of numbers above 1 negative.

Log worked example

log₅(36): no calculator key does base 5, so use change of base. ln(36) = 3.583519 and ln(5) = 1.609438, and 3.583519 ÷ 1.609438 = 2.226566. Check: 5 to the power 2.226566 is 36. The same figure comes from base-10 logs, log(36) ÷ log(5) = 1.556303 ÷ 0.698970, which is the OpenStax worked example rounded to 2.2266.

More Log examples

log₁₀(0.001) = −3, because 10⁻³ = 0.001. pH works this way: a hydrogen-ion concentration of 0.001 mol/L has pH −log₁₀(0.001) = 3.

How many bits does it take to count to a million? log₂(1,000,000) = 19.931569, so 20 bits are needed: 2²⁰ = 1,048,576 is the first power of 2 past a million.

Where the Log calculator can give the wrong answer

The number must be positive. log(0) and log of a negative number are not real numbers, so the page refuses them instead of showing NaN — though the logarithm of a number between 0 and 1 is fine, it is just negative: log₁₀(0.05) = −1.30103.
The base must be positive and not 1: 1 raised to any power is 1, so log base 1 has no answer. A base between 0 and 1 works and gives negative logs for numbers above 1: log₀.₅(8) = −3 because 0.5⁻³ = 8.
Results are shown to six decimal places. A logarithm that should be a whole number is snapped to it when the base's power really is the number; other answers are decimals rounded at the sixth place, so the check line can differ from the input by a unit in the last place.
The natural log row is base e = 2.718281828…, not 'e' in the sense of the exponent key. Choosing base e as the log base gives the same figure as the ln row, which is the point.
This is a real-number logarithm. Complex logarithms of negative numbers exist in higher mathematics and are not handled here.

Spotted a mistake? Tell us

Log FAQ

: For a base's exact powers, count: log₁₀(1000) is 3 because 10 × 10 × 10 = 1000. For anything else, bracket it — log₁₀(50) is between log₁₀(10) = 1 and log₁₀(100) = 2, closer to 2 — then use known values: log₁₀(50) = log₁₀(100) − log₁₀(2) = 2 − 0.30103 = 1.69897. With a calculator that has only ln or log keys, divide: log_b(x) = ln(x) ÷ ln(b).
: ln is the natural logarithm, base e (about 2.71828); log written without a base usually means base 10 in school and engineering, though in some maths and computing texts a bare log means base e or base 2. They differ by a constant factor: ln(x) = 2.302585 × log₁₀(x). The page shows both for every input so there is no guessing.
: Choose 'Any other base', type the base, and the page divides ln(number) by ln(base). log₃(81) = ln(81) ÷ ln(3) = 4.394449 ÷ 1.098612 = 4, because 3⁴ = 81.
: The antilog undoes the logarithm: if log₁₀(x) = 2.5 then x = 10^2.5 = 316.23. The check row on this page does exactly that for your answer, and the exponent calculator on this site raises any base to any power.
: Base 2 counts doublings and bits: log₂(1024) = 10 says 1024 is ten doublings of 1, and a number needs about log₂(n) bits to store. It appears in information theory, algorithm analysis and music (octaves are doublings of frequency).