Log Calculator Online Free Tool
Logarithm Calculator
Logarithm Equation
logb(x) = y ⟺ by = x
Provide any 2 values to calculate the 3rd
Logarithm Calculator
Special inputs: Use "e" for Euler's number (≈2.71828)
Must be positive
Can be any real number
Understanding Logarithms: A Complete Guide
What Are Logarithms?
A logarithm answers the question: "To what power must we raise the base to get the argument?" The logarithm equation logb(x) = y is equivalent to the exponential form by = x. This inverse relationship between logarithms and exponents makes logarithms powerful tools for solving exponential equations and analyzing exponential growth or decay. The base b must be positive and not equal to one, while the argument x must be positive. Understanding this fundamental relationship is essential for applications in science, engineering, finance, and computer science.
Special Logarithm Bases
Natural Logarithm (ln or loge)
Uses Euler's number e ≈ 2.71828 as the base. Natural logarithms are fundamental in calculus because the derivative of ln(x) is simply 1/x. They appear in continuous compounding interest, radioactive decay, population growth models, and probability distributions like the normal distribution. In physics, natural logarithms describe processes that change continuously over time.
Common Logarithm (log or log10)
Uses base 10, matching our decimal number system. Common logarithms measure orders of magnitude and are used in pH calculations (pH = -log[H⁺]), decibel scales for sound intensity, the Richter scale for earthquake magnitude, and astronomical brightness measurements. A difference of 1 in common logarithm represents a tenfold change in the original quantity.
Binary Logarithm (log2)
Uses base 2, essential in computer science and information theory. Binary logarithms calculate the number of bits needed to represent information, analyze algorithm complexity (like binary search with O(log n) time), and determine tree heights in data structures. If log2(n) = k, then n = 2k, representing k binary decisions or halvings.
Real-World Applications
Chemistry and Biology
pH scale measures acidity using -log[H⁺], where neutral water has pH 7 ([H⁺] = 10-7 M). The Henderson-Hasselbalch equation uses logarithms to relate pH to buffer concentrations. Enzyme kinetics and bacterial growth curves involve logarithmic relationships, with doubling times calculated using log2.
Finance and Economics
Continuous compounding uses the formula A = Pert, solved for time using natural logarithms: t = ln(A/P)/r. Investment doubling time follows the Rule of 72, derived from logarithms. Economic growth rates, inflation calculations, and compound annual growth rate (CAGR) all involve logarithmic functions.
Physics and Engineering
Decibel scale for sound: dB = 10·log10(I/I₀), where I₀ is reference intensity. Richter scale for earthquakes: M = log10(A/A₀). Radioactive half-life calculations use ln(2)/λ. Signal processing, electrical engineering, and acoustics rely heavily on logarithmic scales to handle vast ranges of values.
Computer Science
Algorithm complexity analysis uses logarithms: binary search is O(log n), merge sort is O(n log n). Binary trees have maximum height log2(n+1). Information entropy H = -Σp·log2(p) measures data compression limits. Hash table sizing and cryptography implementations depend on logarithmic calculations.
Essential Logarithm Laws
Product Rule: logb(xy) = logb(x) + logb(y)
Quotient Rule: logb(x/y) = logb(x) - logb(y)
Power Rule: logb(xn) = n·logb(x)
Change of Base: logb(x) = loga(x) / loga(b)
Identity: logb(b) = 1
Zero: logb(1) = 0
Inverse: blogb(x) = x
These laws transform complex calculations into simpler operations. The change of base formula lets you calculate any logarithm using available functions like ln or log10. The product rule converts multiplication into addition, which is why logarithmic scales compress large ranges effectively.
Common Mistakes to Avoid
- ✗Wrong: log(x+y) = log(x) + log(y). Correct: log(xy) = log(x) + log(y)
- ✗Domain error: Never take logarithms of negative numbers or zero in real number systems
- ✗Base confusion: log(x) typically means log10(x), while ln(x) always means loge(x)
- ✓Check domain: Always verify x > 0 and b > 0, b ≠ 1 before calculating
- ✓Use calculator efficiently: Most calculators only have ln and log buttons; use change of base formula for others
Mastering logarithms opens doors to understanding exponential relationships throughout science and mathematics. Whether you're analyzing earthquake magnitudes, calculating investment returns, optimizing algorithms, or measuring chemical concentrations, logarithms provide the mathematical framework to work with quantities spanning many orders of magnitude.
This calculator supports all three modes of logarithm problems: finding the result when base and argument are known, finding the argument (anti-logarithm) when base and result are known, or finding the base when argument and result are known. Practice with different bases and values to develop intuition for how logarithms behave and transform numbers.