Logarithm Calculator | Log, Ln, Lg & Any Base Logarithm

📖 How to Use This Logarithm Calculator

Select logarithm type - Choose Custom Base, Natural Log (ln), Common Log (lg), or Log Base 2.

Enter the base (for custom log) - Input the base of the logarithm (must be >0 and ≠1).

Enter the number (x) - Input the number you want to find the logarithm of (must be >0).

Click "Calculate" - Get the result with step-by-step explanation.

📐 What is a Logarithm?

A logarithm answers the question: "To what exponent must the base be raised to get a given number?"

The logarithm of x with base b is written as logₐ(x) = y, which means aʸ = x. For example, log₁₀(100) = 2 because 10² = 100.

📊 Common Logarithm Types

Common Logarithm (lg / log₁₀): Base 10. Used in scientific notation, decibels, pH scale.
Natural Logarithm (ln): Base e (≈ 2.71828). Used in calculus, exponential growth/decay, compound interest.
Binary Logarithm (lb / log₂): Base 2. Used in computer science, information theory, data structures.
Custom Base: Any positive number ≠ 1. Used in specialized applications.

🧮 Logarithm Rules and Properties

Product Rule: logₐ(m × n) = logₐ(m) + logₐ(n)
Quotient Rule: logₐ(m / n) = logₐ(m) - logₐ(n)
Power Rule: logₐ(mⁿ) = n × logₐ(m)
Change of Base: logₐ(b) = log_c(b) / log_c(a)
Special Values: logₐ(1) = 0, logₐ(a) = 1

💡 Real-World Applications of Logarithms

Richter Scale (Earthquakes): Logarithmic scale where each whole number increase represents 10x amplitude
Decibels (Sound): Logarithmic scale for sound intensity and power
pH Scale: Measures acidity as negative logarithm of hydrogen ion concentration
Compound Interest: Calculating time to reach investment goals
Algorithm Complexity: Binary search, merge sort use log₂ complexity

❓ Frequently Asked Questions (FAQ)

What is the difference between log and ln?

Log typically means log₁₀ (common logarithm) while ln means natural logarithm (base e ≈ 2.71828). Otherwise, they follow the same rules.

Why can't a logarithm have a negative base?

Logarithms with negative bases are not defined in real numbers because they produce complex results. For real-valued logarithms, the base must be positive and not equal to 1.

What is the log of 0?

logₐ(0) is undefined (approaches -∞). There's no exponent that gives zero because a positive base raised to any power is always positive.

What is the difference between log and antilog?

Logarithm finds the exponent; antilog is the inverse operation: if y = logₐ(x), then x = antilogₐ(y) = aʸ.

How do calculators compute logarithms?

Calculators use series expansions (Taylor series) or the change of base formula with precomputed values to compute logarithms accurately.

What is log base 2 used for in computer science?

Log base 2 appears in algorithm analysis (binary search: O(log₂ n)), data structures, and information theory (bits needed to represent n values).