Logarithm Calculator - Log Base 10, ln & Log Base 2

What Is a Logarithm?

A logarithm answers one question: what power must you raise a base to in order to reach a given number? This logarithm calculator computes the three bases you meet most often — the common log (base 10), the natural log (ln, base e), and the binary log (base 2) — the instant you type. Enter any positive value, pick a base, and read a high-precision result with no submit step.

Definition: the logarithm is the inverse of exponentiation. If bˣ = y, then log_b(y) = x. For example, log₁₀(100) = 2 because 10² = 100.

The Three Bases

log₁₀ — Common Log

Base 10, used in scientific scales such as pH, decibels, and the Richter scale.

ln — Natural Log

Base e (≈ 2.718), the backbone of calculus, growth and decay, and continuous compounding.

log₂ — Binary Log

Base 2, central to computer science, information theory, and algorithm complexity.

How to Calculate a Logarithm

Choose the Base

Pick log₁₀, ln, or log₂ from the type selector. The formula line updates to match your choice, so you always see the expression you are solving.

Enter a Positive Number

Type any value greater than zero. Logarithms are undefined for zero and negative numbers, so only positive inputs return a result.

Read the Result

The answer appears instantly as you type, shown to up to 10 decimal places with trailing zeros trimmed for a clean value.

Positive inputs only: the log of zero tends to negative infinity, and the log of a negative number is undefined in the real numbers.

Private by design: every calculation runs in your browser — nothing you type is uploaded to a server.

Features

Three Bases in One

Switch freely between log₁₀, ln, and log₂ without leaving the page.

Real-Time Calculation

Results update the instant you type — no button to press.

Live Formula Preview

The expression line (for example log₁₀(100) =) mirrors your base and input.

High Precision

Up to 10 decimal places, accurate for round numbers and awkward decimals alike.

Common Logarithm Values

Expression	Result	Why
log₁₀(10)	1	10¹ = 10
log₁₀(100)	2	10² = 100
ln(e)	1	e¹ = e
ln(1)	0	e⁰ = 1
log₂(2)	1	2¹ = 2
log₂(8)	3	2³ = 8

Frequently Asked Questions

How do I calculate a logarithm?

Choose a base (log₁₀, ln, or log₂), enter a positive number, and the result appears instantly. Mathematically, log_b(y) asks "to what power must I raise b to get y?" — so log₁₀(100) = 2 because 10² = 100.

What is the difference between log, ln, and log₂?

They differ only in the base. "log" usually means the common logarithm with base 10, "ln" is the natural logarithm with base e (≈ 2.718), and "log₂" is the binary logarithm with base 2. The same input gives a different value for each base.

Can you take the log of a negative number or zero?

No. For any positive base, raising it to a real power always yields a positive result, so no real exponent produces zero or a negative number. That is why this calculator accepts positive inputs only and flags anything else.

How does the change-of-base formula work?

To find a logarithm in any base b, divide two logs that share a common base: log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b). It lets you compute an unusual base from the common or natural log this tool already provides.

What is log base 2 used for?

The binary logarithm counts how many times you can halve a value. It measures bits of information and expresses the complexity of algorithms such as binary search and merge sort, which is why it appears throughout computer science.