Log Calculator

A logarithm is the inverse operation of exponentiation. It answers the question: "To what power must we raise a base number to get a specific result?" For example, log₁₀(100) = 2 because 10² = 100. Logarithms are fundamental in mathematics and appear in many real-world applications including compound interest calculations, pH scales in chemistry, and the Richter scale for earthquakes.

To calculate log base 10 (written as log₁₀ or simply "log"), use our calculator above by selecting "Common Log (log₁₀)" and entering your number. The common logarithm tells you how many times you need to multiply 10 by itself to get your number. For example, log₁₀(1000) = 3 because 10 × 10 × 10 = 1000. Common logarithms are particularly useful in scientific notation and engineering applications.

The main difference is the base: natural logarithm (ln) uses base e (approximately 2.71828), while common logarithm (log) uses base 10. Natural logarithms appear frequently in calculus, exponential growth and decay (like in radioactive decay calculations), and continuous compound interest. Common logarithms are more intuitive for everyday calculations and are widely used in engineering, pH calculations, and decibel measurements.

Key logarithm properties include: (1) Product Rule: log(xy) = log(x) + log(y), (2) Quotient Rule: log(x/y) = log(x) - log(y), (3) Power Rule: log(xⁿ) = n·log(x), (4) log(1) = 0 for any base, and (5) log_b(b) = 1. These rules allow you to simplify complex logarithmic expressions and solve exponential equations. Understanding these properties is essential for advanced mathematics and scientific applications.

Logarithms are used extensively in science and engineering: measuring earthquake intensity (Richter scale), sound levels (decibels), acidity (pH scale), star brightness, and data compression. In finance, they help calculate compound interest and investment growth. Computer science uses binary logarithms for algorithm complexity analysis. In chemistry and physics, they model radioactive decay, bacterial growth, and other exponential processes. You can explore related calculations using our percentage calculator for growth rates.

You can convert between logarithm bases using the change of base formula: log_b(x) = log_a(x) / log_a(b), where 'a' is any positive base. For example, to convert log₂(8) to base 10: log₂(8) = log₁₀(8) / log₁₀(2) ≈ 0.903 / 0.301 = 3. This formula is built into our calculator when you select "Custom Base" - simply enter your desired base and the calculator handles the conversion automatically.

An antilog (antilogarithm) is the inverse operation of a logarithm - it "undoes" what the logarithm does. If log_b(x) = y, then antilog_b(y) = x, which equals b^y. For example, if log₁₀(100) = 2, then antilog₁₀(2) = 10² = 100. Use the Antilog Calculator section above to quickly compute antilogarithms. This is particularly useful when working backwards from logarithmic scales or when solving exponential equations.

In real numbers, logarithms are only defined for positive numbers. This is because no real power of a positive base can produce zero or a negative result. For example, 10^x will always be positive no matter what real number x is. As x approaches negative infinity, 10^x approaches zero but never reaches it, which is why log(0) is undefined (or negative infinity in limits). While complex number systems can handle logarithms of negative numbers, standard calculators work with real numbers only.

Binary logarithms (log₂) are essential in computer science and information theory. Use log₂ when working with: binary data and bit calculations, algorithm complexity (like binary search which runs in O(log₂ n) time), determining how many times you can halve a number, calculating the number of bits needed to represent a number, or analyzing binary trees and data structures. For example, log₂(1024) = 10 tells you that 1024 = 2¹⁰, useful for understanding data sizes in computing.

Logarithms are crucial for solving exponential growth and decay problems. When you have an equation like A = A₀e^(kt) and need to find time 't', you use natural logarithms: t = ln(A/A₀) / k. This applies to population growth, radioactive decay (see our half-life calculator), compound interest, bacterial cultures, and carbon dating. The doubling time or half-life of a quantity can be found using t = ln(2) / k ≈ 0.693 / k, making logarithms indispensable for these calculations.