Logarithm Calculator

Calculate logarithms, antilogarithms, natural logs, exponential functions and solve logarithmic equations with step-by-step solutions and multiple base support.

Basic Logarithm Calculator

Calculate logarithms and antilogarithms with any base.

Calculation Type

Value (b)

Base (a)

Logarithm: logₐb = x means aˣ = b
Antilogarithm: antilogₐ(b) = aᵇ

Common Logarithm Values

• log₁₀(10) = 1 • log₁₀(100) = 2 • log₁₀(1000) = 3
• log₂(8) = 3 • log₂(16) = 4 • log₂(32) = 5
• ln(e) = 1 • ln(e²) = 2 • ln(e³) = 3

log₍10₎(100)

Logarithm Result

2.000000

Calculation:
log₍10₎(100) = x
10ˣ = 100
x = log(100) ÷ log(10)
x = 2.000000

Equivalent Expressions:

Natural log: 0.693147

Common log: 0.301030

Logarithm Guide

Logarithm Basics

Definition: logₐb = x means aˣ = b
The logarithm is the inverse operation of exponentiation.

Key Formulas

Product Rule: logₐ(mn) = logₐm + logₐn
Quotient Rule: logₐ(m/n) = logₐm - logₐn
Power Rule: logₐ(mⁿ) = n·logₐm
Change of Base: logₐb = logₓb / logₓa
Natural Log: ln(x) = logₑ(x)

Common Logarithm Types

• Common Logarithm: log₁₀(x) - Base 10
• Natural Logarithm: ln(x) - Base e (≈2.718)
• Binary Logarithm: log₂(x) - Base 2
• Arbitrary Base: logₐ(x) - Any positive a ≠ 1

Real-World Applications

• Science: pH scale, Richter scale, decibels
• Finance: Compound interest, exponential growth
• Computer Science: Algorithm complexity, information theory
• Engineering: Signal processing, control systems
• Statistics: Data normalization, probability

Important Logarithmic Properties

logₐ1 = 0

Any base raised to 0 equals 1

logₐa = 1

Base raised to 1 equals itself

logₐaˣ = x

Logarithm and exponential cancel

a^(logₐx) = x

Exponential and logarithm cancel

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