Logarithm Calculator (log, ln, log₂)

How to use this calculator

Enter the base and the number to get the logarithm. To go backwards (antilog), press × on Number and type the logarithm.

Common bases: 10 (common log), 2 (binary log), 2.718281828 (natural log, e).

The formula

log_b(x) = y ⇔ bʸ = x
log_b(x) = ln(x) ÷ ln(b)

Examples

Expression	Value	Because
log₁₀ 1000	3	10³ = 1000
log₂ 64	6	2⁶ = 64
ln 20	2.996	e^2.996 ≈ 20
log₁₀ 0.01	−2	10⁻² = 0.01

Rules of logarithms

• log(a × b) = log a + log b
• log(a ÷ b) = log a − log b
• log(aⁿ) = n × log a
• log_b(1) = 0 for any base

Where logs are used

• Decibels: sound level = 10 × log₁₀(P ÷ P₀).
• pH: pH = −log₁₀[H⁺].
• Richter scale: each step is 10× larger in amplitude.
• Doubling time: years to double = ln 2 ÷ ln(1 + rate).

Frequently asked questions

What is a logarithm?

The logarithm tells you what power the base must be raised to to get the number. log₁₀ 1000 = 3 because 10³ = 1000.

How do I calculate the natural log (ln)?

Use base e ≈ 2.718281828. Enter 2.718281828 as the base. ln 20 ≈ 2.996.

How do I find the antilog?

Press × on Number, then enter the base and the logarithm. The antilog of 2 in base 10 is 100.

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