Calculate logarithms with any base
Your calculations will appear here
Step 1: We want to find log10(100)
Step 2: 102 = 100
Step 3: Therefore, log10(100) = 2
A logarithm answers the question: "To what exponent must we raise the base to get a certain number?"
If \( b^x = y \), then \( \log_b(y) = x \)
Logarithm with base 10, written as log(x) or log₁₀(x)
Logarithm with base e (≈2.718), written as ln(x)
Our free Logarithm Calculator is an essential mathematical tool for students, engineers, scientists, and professionals working with exponential relationships. Whether you're solving logarithmic equations, analyzing exponential growth, calculating pH in chemistry, working with decibels in physics, or solving complex mathematical problems, this calculator provides precise log and natural log calculations.
Calculate common logarithms (log₁₀), natural logarithms (ln), logarithms with any base, perform logarithmic transformations, and solve exponential equations with our specialized mathematics tool. Perfect for algebra, calculus, statistics, physics, chemistry, and engineering applications.
Calculate logarithms with any base: common base 10 (log₁₀), natural base e (ln), binary base 2 (log₂), and custom bases. Switch between bases instantly for different applications.
High-precision calculations with up to 15 decimal places accuracy using advanced algorithms. Handles very large numbers (up to 10³⁰⁰) and very small numbers (down to 10⁻³⁰⁰).
Step-by-step solutions showing logarithmic properties, change of base formula applications, and relationship between logarithmic and exponential forms. Perfect for learning and teaching.
Useful for scientific calculations: pH in chemistry, Richter scale in geology, decibels in acoustics, exponential growth in biology, and complex algorithms in computer science.
Used by students, teachers, engineers, scientists, and researchers worldwide. No registration required - calculate logarithms instantly for academic, professional, or personal use!
log (or log₁₀) is the common logarithm with base 10, commonly used in engineering and science. ln is the natural logarithm with base e (approximately 2.71828), used extensively in mathematics, physics, and calculus. The relationship is: ln(x) = log(x) / log(e) ≈ 2.302585 × log(x).
The logarithm of a negative real number is undefined in real numbers but defined in complex numbers. In complex analysis, ln(-x) = ln(x) + iπ (where i is the imaginary unit). Our calculator handles complex logarithm calculations with appropriate notation.
The change of base formula allows converting between different bases: logₐ(x) = logₙ(x) / logₙ(a), where n can be any positive number (usually 10 or e). For example: log₂(8) = log₁₀(8) / log₁₀(2) = 0.90309 / 0.30103 = 3.
Common applications include: pH calculation (pH = -log[H⁺]), earthquake magnitude (Richter scale), sound intensity (decibels), compound interest calculations, radioactive decay, algorithm complexity (Big O notation), and data compression.