Logarithm Calculator

Calculate natural logarithm (ln), common logarithm (log₁₀), and logarithms with custom bases

💡 Natural logarithm uses base e ≈ 2.71828

Natural Logarithm Result

Natural Log Formula: ln(x) = log_e(x)
Inverse (Antilog): e^ln(x) = x
Domain: x > 0
💡 Common logarithm uses base 10

Common Logarithm Result

Common Log Formula: log(x) = log₁₀(x)
Inverse (Antilog): 10^log(x) = x
Domain: x > 0
💡 Base must be positive and not equal to 1

Custom Base Logarithm

Custom Log Formula: log_b(x) = ln(x)/ln(b)
Inverse (Antilog): b^log_b(x) = x
Domain: x > 0, b > 0, b ≠ 1
💡 Antilog is the inverse of logarithm (b^x)

Antilogarithm Result

Antilog Formula: antilog_b(x) = b^x
Inverse: log_b(b^x) = x
Domain: All real numbers
Logarithm Properties & Rules
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)
Identity: log_b(b) = 1, log_b(1) = 0

How to Use the Logarithm Calculator

Our comprehensive logarithm calculator supports natural logarithms, common logarithms, custom bases, and antilogarithms with high precision:

📊 Natural Logarithm (ln)

Natural logarithm uses Euler's number e ≈ 2.71828 as the base. Enter any positive number to calculate ln(x). Commonly used in calculus, physics, and engineering for exponential growth and decay problems.

🔢 Common Logarithm (log₁₀)

Common logarithm uses base 10, the foundation of our decimal number system. Perfect for scientific notation, pH calculations, decibel levels, and Richter scale measurements.

⚙️ Custom Base Logarithm

Calculate logarithms with any positive base (except 1). Enter both the base and the number. Uses the change of base formula: log_b(x) = ln(x)/ln(b). Essential for computer science (binary logs) and advanced mathematics.

🔄 Antilogarithm

Antilog is the inverse of logarithm, calculated as b^x. Enter the base and the logarithm value to find the original number. Useful for converting back from logarithmic scales.

📐 Mathematical Properties

The calculator includes essential logarithm properties:
• Product Rule: log(xy) = log(x) + log(y)
• Quotient Rule: log(x/y) = log(x) - log(y)
• Power Rule: log(x^n) = n × log(x)
• Change of Base: Convert between different logarithm bases

💡 Practical Applications

• Science: pH calculations, earthquake measurements, sound levels
• Finance: Compound interest, investment growth calculations
• Engineering: Signal processing, exponential decay analysis
• Computer Science: Algorithm complexity, binary logarithms
• Statistics: Log-normal distributions, data transformations

🔧 Pro Tips

• Input must be positive (logarithm domain requirement)
• Base must be positive and not equal to 1
• Results are accurate to 10+ decimal places
• Use e ≈ 2.71828 for natural logarithm base
• Verify results using the antilog function

Frequently Asked Questions

"ln" is the natural logarithm with base e ≈ 2.71828, while "log" typically refers to the common logarithm with base 10. In some contexts, "log" might refer to natural logarithm, but our calculator clearly distinguishes between ln (natural) and log₁₀ (common).
Logarithms of negative numbers are not defined in real numbers because no real power of a positive base can produce a negative result. For negative numbers, you would need complex logarithms, which involve imaginary numbers.
Use the change of base formula: log_b(x) = log_c(x) / log_c(b), where c is any convenient base (usually 10 or e). For example, to convert log₂(8) to natural log: ln(8) / ln(2) = 2.079 / 0.693 = 3.
Antilogarithm is the inverse operation of logarithm. If log_b(x) = y, then antilog_b(y) = x, calculated as b^y. Use it when you have a logarithmic value and need to find the original number, such as converting from decibels back to intensity ratios.
Our calculator provides results accurate to 10+ decimal places using JavaScript's built-in mathematical functions. This precision is suitable for most scientific, engineering, and academic applications. For extremely high-precision requirements, specialized mathematical software may be needed.