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Logarithm Calculator

Calculate log with any base

Must be positive

Common Logarithms

Natural Log (ln) - Base e

ln(1):0.000
ln(2):0.693
ln(10):2.303
ln(100):4.605
ln(1000):6.908

Common Log - Base 10

log(1):0.0
log(10):1.0
log(100):2.0
log(1000):3.0
log(10000):4.0

Binary Log - Base 2

log₂(1):0
log₂(2):1
log₂(4):2
log₂(8):3
log₂(16):4
log₂(32):5
log₂(64):6
log₂(128):7

What is a Logarithm?

A logarithm answers the question: "What power do I raise the base to, to get this number?"

Formula

If b^y = x, then logb(x) = y

Logarithms are the inverse of exponentiation

Example

log₁₀(100) = 2 because 10² = 100

  • Base: 10
  • Number: 100
  • Answer: 2 (10 raised to power 2 equals 100)

Logarithm Rules

log(xy) = log(x) + log(y)
Product rule
Example: log(20) = log(4×5) = log(4) + log(5)
log(x/y) = log(x) - log(y)
Quotient rule
Example: log(5) = log(10/2) = log(10) - log(2)
log(x^n) = n × log(x)
Power rule
Example: log(100) = log(10²) = 2×log(10)
log_b(b) = 1
Log of base equals 1
Example: log₁₀(10) = 1
log_b(1) = 0
Log of 1 equals 0
Example: log(1) = 0
log_b(b^x) = x
Log and exponent cancel
Example: log(10³) = 3
b^(log_b(x)) = x
Exponent and log cancel
Example: 10^(log(5)) = 5

Types of Logarithms

Natural Logarithm (ln)

Base e (≈ 2.71828). Written as ln(x) or loge(x)

Most common in calculus, science, and continuous growth models

Common Logarithm (log)

Base 10. Written as log(x) or log₁₀(x)

Used in pH scale, decibels, Richter scale, and everyday calculations

Binary Logarithm (log₂)

Base 2. Written as log₂(x) or lb(x)

Used in computer science, information theory, and binary systems

Real-World Applications

🔬 Science

  • • pH scale (acidity): pH = -log[H⁺]
  • • Richter scale (earthquakes)
  • • Decibels (sound intensity)
  • • Half-life calculations

💻 Computer Science

  • • Algorithm complexity (Big O)
  • • Binary search trees
  • • Information theory
  • • Data compression

💰 Finance

  • • Compound interest calculations
  • • Time to double money
  • • Growth rate analysis
  • • Investment returns

📊 Statistics

  • • Log-normal distributions
  • • Data transformation
  • • Regression analysis
  • • Scale normalization

How to Use This Calculator

1

Enter the Number

Input the positive number (x) for which you want to find the logarithm. The number must be greater than 0.

2

Choose or Enter Base

Select a common base (2, e, or 10) using the quick buttons, or enter any positive base (not equal to 1) manually.

3

Calculate

Click "Calculate Logarithm" to get the logarithm result, along with natural log, common log, binary log, and antilog for verification.

Formula

If by = x, then logb(x) = y

Where:

  • b = base (positive number, not equal to 1)
  • x = number (must be positive)
  • y = logarithm result

Example 1: log₁₀(100)

10y = 100

10² = 100

Therefore: log₁₀(100) = 2

Example 2: log₂(8)

2y = 8

2³ = 8

Therefore: log₂(8) = 3

Example 3: ln(e)

ey = e

e¹ = e

Therefore: ln(e) = 1

About Log Calculator

The Log Calculator (Logarithm Calculator) helps you calculate logarithms with any base. A logarithm answers the question: "What power do I raise the base to, to get this number?" This calculator supports common logarithm (base 10), natural logarithm (base e), binary logarithm (base 2), and any other base you specify. It also shows antilog for verification.

When to Use This Calculator

  • Mathematics: Calculate logarithms for algebra, calculus, and advanced math problems
  • Chemistry: Calculate pH values: pH = -log[H⁺]
  • Physics: Work with decibels, Richter scale, or exponential decay
  • Computer Science: Calculate binary logarithms for algorithm complexity analysis
  • Finance: Calculate time to double investments or compound interest problems

Why Use Our Calculator?

  • ✅ Multiple Bases: Calculate logarithms with base 2, e, 10, or any custom base
  • ✅ All Log Types: Shows natural log (ln), common log (log₁₀), and binary log (log₂) simultaneously
  • ✅ Verification: Includes antilog to verify your result
  • ✅ Educational: Learn logarithm rules and properties
  • ✅ 100% Free: No registration or payment required
  • ✅ Accurate: High precision calculations

Common Applications

Chemistry - pH Scale: If hydrogen ion concentration [H⁺] = 0.001 M, then pH = -log₁₀(0.001) = -log₁₀(10⁻³) = -(-3) = 3.

Sound - Decibels: Sound intensity level in decibels = 10 × log₁₀(I/I₀), where I is the intensity and I₀ is the reference intensity.

Computer Science - Binary Search: The maximum number of comparisons needed to find an item in a sorted array of size n is log₂(n).

Tips for Best Results

  • The number (x) must be positive - logarithms of negative numbers are not real numbers
  • The base (b) must be positive and not equal to 1
  • Use quick buttons for common bases (2, e, 10) for convenience
  • Check the antilog result to verify your calculation: it should equal your input number
  • Remember: log(1) = 0 and log(base) = 1 for any base

Frequently Asked Questions

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

For real numbers, you can't raise a positive base to any power and get a negative result. log(-5) has no real answer. (Complex numbers have a solution, but that's advanced math!)

What's the difference between ln and log?

ln is natural log (base e ≈ 2.718). log usually means base 10. Both follow the same rules, just different bases. ln(x) = log e(x).

Why is log(1) always 0?

Any number raised to power 0 equals 1. So b^0 = 1, which means log b(1) = 0. Works for any base!

How do I calculate log with a different base?

Use the change of base formula: log b(x) = log(x) / log(b) or ln(x) / ln(b). This lets you use base 10 or e to calculate any base.

What's an antilog?

Antilog is the inverse of log. If log b(x) = y, then antilog b(y) = x. It's the same as exponentiation: antilog = b^y.