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Logarithm Calculator

Calculate logarithms with any base, including natural logarithm (ln), common logarithm (log₁₀), and binary logarithm (log₂). Perfect for algebra, calculus, and scientific calculations.

Calculate Logarithm

Quick Examples

Logarithm Properties

Product Rule:
log_b(x × y) = log_b(x) + log_b(y)
Quotient Rule:
log_b(x ÷ y) = log_b(x) - log_b(y)
Power Rule:
log_b(x^n) = n × log_b(x)
Change of Base:
log_b(x) = log_c(x) ÷ log_c(b)

Calculation Results

Logarithm Result:
2.000000
Expression:
log_10(100)
Decimal Places:
2.0000000000
Verification:
10.0000^2.000000 = 100.000000

Alternative Representations

Common Logarithm (log₁₀):
2.000000
Natural Logarithm (ln):
4.605170
Binary Logarithm (log₂):
6.643856

Step-by-Step Calculation

Step 1: Identify the logarithm
log_10(100)
Step 3: Calculate the result
Result = 2.000000

How to Use This Calculator

This logarithm calculator helps you compute logarithms with various bases and explore logarithmic properties.

  1. Select your calculation type from the dropdown menu
  2. Choose the appropriate base (common, natural, binary, or custom)
  3. Enter the required values based on your calculation type
  4. View the calculated result and step-by-step solution
  5. Explore alternative representations and verification

Understanding Logarithms

What is a Logarithm?

A logarithm is the inverse operation of exponentiation. If b^y = x, then log_b(x) = y. The logarithm answers the question: "To what power must we raise the base to get this number?"

Common Types of Logarithms

  • Common Logarithm (log₁₀): Base 10, often written as "log"
  • Natural Logarithm (ln): Base e (≈2.718), used in calculus and natural sciences
  • Binary Logarithm (log₂): Base 2, used in computer science and information theory
  • Custom Base: Any positive number except 1

Logarithm Properties

  • Product Rule: log_b(xy) = log_b(x) + log_b(y)
  • Quotient Rule: log_b(x/y) = log_b(x) - log_b(y)
  • Power Rule: log_b(x^n) = n × log_b(x)
  • Change of Base: log_b(x) = log_c(x) / log_c(b)

Special Values

  • log_b(1) = 0 for any base b
  • log_b(b) = 1 for any base b
  • log_b(b^n) = n for any base b
  • b^(log_b(x)) = x for any base b

Example Calculations

Example 1: Common Logarithm

Calculate log₁₀(1000)

Solution: Since 10³ = 1000, log₁₀(1000) = 3

Example 2: Natural Logarithm

Calculate ln(e²)

Solution: Since e² = e², ln(e²) = 2

Example 3: Change of Base

Convert log₅(25) to base 10

Solution: log₅(25) = log₁₀(25) / log₁₀(5) = 1.398 / 0.699 = 2

Example 4: Logarithm Properties

Simplify log₂(8 × 4) using the product rule

Solution: log₂(8 × 4) = log₂(8) + log₂(4) = 3 + 2 = 5

Frequently Asked Questions

Can you take the logarithm of a negative number?

No, you cannot take the logarithm of a negative number or zero in the real number system. The argument of a logarithm must be positive. However, complex logarithms can handle negative numbers.

What's the difference between log and ln?

"log" typically refers to the common logarithm (base 10), while "ln" refers to the natural logarithm (base e). In some contexts, "log" might refer to logarithms in general or even natural logarithms, so it's important to check the context.

Why can't the base be 1?

If the base were 1, then 1^x = 1 for any value of x, making it impossible to determine a unique logarithm value. The logarithm would be undefined because there's no unique answer to "1 to what power equals this number?"

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