Logarithm Calculator - Log Base 10, Ln, Log2, Custom Base

How the Logarithm Calculator Works

Enter a number and choose the logarithm type. The calculator computes the result using the change-of-base formula and shows a step-by-step breakdown. Supports common (log10), natural (ln), binary (log2), and any custom base you need.

Logarithm Types - Quick Reference

Four types of logarithms are used in mathematics, science and engineering. Each has a specific base and application domain:

Type	Notation	Base	Used In
Common	log(x)	10	Science, engineering, pH scale, decibels
Natural	ln(x)	e (2.71828...)	Calculus, statistics, physics, finance
Binary	log2(x)	2	Computer science, information theory, data storage
Custom base	log_a(x)	any a > 0, a != 1	Specific math problems, music theory

Key Logarithm Properties

These identities simplify logarithmic calculations and are essential for algebra and calculus:

Property	Formula	Example
Product rule	*log(a b) = log(a) + log(b)**	log(20) = log(4) + log(5) = 0.602 + 0.699 = 1.301
Quotient rule	log(a / b) = log(a) - log(b)	log(5) = log(10) - log(2) = 1 - 0.301 = 0.699
Power rule	*log(a^n) = n log(a)**	log(1000) = 3 * log(10) = 3 * 1 = 3
Change of base	log_a(x) = log(x) / log(a)	log_5(125) = log(125) / log(5) = 2.097 / 0.699 = 3
Log of 1	log_a(1) = 0 for any base	log(1) = 0, ln(1) = 0, log2(1) = 0
Log of the base	log_a(a) = 1	log(10) = 1, ln(e) = 1, log2(2) = 1

Practical Examples

Example 1: log(1000) = 3 because 10^3 = 1000

Example 2: ln(e^4) = 4 because the natural log and exponential cancel out

Example 3: log2(256) = 8 because 2^8 = 256 (256 bytes = 8 bits)

Example 4: log5(125) = 3 because 5^3 = 125

Example 5: log(0.01) = -2 because 10^(-2) = 0.01

Example 6: ln(7.389) = 2 because e^2 = 7.389

Example 7: log2(1024) = 10 because 2^10 = 1024 (1 KB in computing)

Real-World Applications of Logarithms

Field	Application	Log Type
Seismology	Richter scale - each whole number is 10x more energy	log10
Sound	Decibels (dB) - 20 dB increase = 10x louder	log10
Chemistry	pH = -log[H+] - measures acidity (pH 3 is 100x more acidic than pH 5)	log10
Finance	Compound interest doubling time: t = ln(2) / ln(1+r)	ln
Computer Science	Binary search complexity O(log2 n), bits needed for n values	log2
Information Theory	Shannon entropy H = -sum(p * log2(p)) - measures information content	log2

FAQ

What is a logarithm?

A logarithm answers the question: to what power must the base be raised to produce a given number? For example, log10(100) = 2 because 10^2 = 100. In other words, log_b(x) = y means b^y = x.

Why can't I take the log of zero or a negative number?

In the real number system, logarithms are only defined for positive numbers. No real power of a positive base can produce zero or a negative result. log(0) approaches negative infinity. Complex logarithms exist for negative numbers but are beyond basic calculation.

What is the change-of-base formula?

log_a(x) = ln(x) / ln(a) = log(x) / log(a). This formula converts any logarithm to a base your calculator supports (usually 10 or e). For example, log_5(125) = log(125) / log(5) = 2.097 / 0.699 = 3.

What is ln (natural logarithm) and why is it important?

ln is the logarithm with base e (Euler's number, approximately 2.71828). It appears naturally in calculus because the derivative of ln(x) is simply 1/x. It models continuous growth and decay: radioactive decay, population growth, compound interest with continuous compounding, and electrical circuits.

What is log2 used for in computer science?

Binary logarithm (log base 2) tells you how many bits are needed to represent a number. log2(256) = 8, meaning 256 values need 8 bits (1 byte). It also defines algorithm complexity: binary search on n items takes log2(n) steps. A sorted list of 1,000,000 items needs only 20 comparisons.

How do I convert between log types?

Use the change-of-base formula: log_a(x) = log_b(x) / log_b(a). Common conversions: ln(x) = log(x) / log(e) = log(x) / 0.4343. Or log(x) = ln(x) / ln(10) = ln(x) / 2.3026. The calculator handles all conversions automatically.

What is the difference between log and ln?

log (without subscript) usually means base 10 in science and engineering, or base e in pure mathematics. ln always means base e (natural logarithm). To avoid confusion, this calculator labels them explicitly: log10 for common and ln for natural.

Can a logarithm be negative?

Yes. The result of a logarithm can be negative when the argument is between 0 and 1. For example, log(0.1) = -1 because 10^(-1) = 0.1. But the argument (input number) must always be positive.

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