Logarithm Calculator
Free online logarithm calculator to compute logarithms of any base, natural logs, common logs, and solve logarithmic equations with step-by-step solutions.
🔢Logarithm Calculator
Calculate logarithms, solve equations, and convert between bases
Calculate log_b(x) for any base b
Logarithm Properties Quick Reference
Basic Properties
log_b(1) = 0log_b(b) = 1log_b(b^x) = xb^(log_b(x)) = x
Logarithm Rules
log_b(xy) = log_b(x) + log_b(y)log_b(x/y) = log_b(x) - log_b(y)log_b(x^n) = n * log_b(x)log_a(x) = log_b(x) / log_b(a)
Related Calculators
About This Calculator
Logarithms are one of the most powerful mathematical tools, serving as the inverse operation of exponentiation. If you understand that 10^2 = 100, then the logarithm tells you the exponent: log₁₀(100) = 2. This fundamental relationship makes logarithms indispensable across science, engineering, finance, and technology. Our logarithm calculator simplifies complex logarithmic computations, supporting any base including natural logarithms (ln) and common logarithms (log₁₀). Whether you need to calculate the pH of a solution, determine earthquake magnitudes on the Richter scale, compute compound interest over time, or solve exponential equations in your math homework, this tool provides accurate results with step-by-step explanations. The calculator handles the change of base formula automatically, converting between different logarithmic bases effortlessly. Understanding logarithms opens doors to comprehending exponential growth and decay, signal processing, information theory, and countless real-world phenomena that follow logarithmic scales. From measuring sound intensity in decibels to calculating the half-life of radioactive materials, logarithms are everywhere in our daily lives. For related calculations, try our Exponent Calculator for power operations or our Scientific Calculator for comprehensive mathematical functions.
How to Use the Logarithm Calculator
- 1Enter values and click calculate
Understanding Logarithms: The Basics
A logarithm answers the question: 'To what power must we raise the base to get a certain number?' Written as log_b(x) = y, this means b^y = x. The base (b) must be positive and not equal to 1, and the argument (x) must be positive. For example, log₂(8) = 3 because 2³ = 8. This inverse relationship with exponentiation makes logarithms essential for solving equations where the unknown is in the exponent. The concept was invented by John Napier in the early 17th century and revolutionized computation before calculators existed, as it converts multiplication into addition and division into subtraction. For calculations involving powers and exponents directly, see our Exponent Calculator.
Natural Logarithms vs. Common Logarithms
The two most frequently used logarithms are natural logarithms (ln) with base e ≈ 2.71828, and common logarithms (log or log₁₀) with base 10. Natural logarithms arise naturally in calculus and are fundamental to continuous growth and decay processes. The constant e appears throughout mathematics, physics, and engineering. Common logarithms are intuitive for base-10 number systems and are widely used in engineering, chemistry, and everyday calculations. When you see 'log' without a base specified, it typically means log₁₀ in applied sciences or ln in pure mathematics.
Key Logarithm Properties and Rules
Mastering logarithm properties simplifies complex calculations. The Product Rule states log_b(xy) = log_b(x) + log_b(y), turning multiplication into addition. The Quotient Rule states log_b(x/y) = log_b(x) - log_b(y), converting division to subtraction. The Power Rule states log_b(x^n) = n × log_b(x), bringing exponents down as multipliers. Additional properties include log_b(1) = 0 for any base, log_b(b) = 1, and the inverse relationship b^(log_b(x)) = x. These rules form the foundation for simplifying logarithmic expressions and solving equations.
The Change of Base Formula
The change of base formula allows you to convert logarithms from one base to another: log_a(x) = log_b(x) / log_b(a). This is how calculators compute logarithms with any base using only natural or common logarithms. For example, to find log₂(10), you can calculate ln(10) / ln(2) ≈ 3.322. This formula is invaluable when your calculator only has ln and log₁₀ buttons but you need a different base. It also proves useful in comparing quantities measured on different logarithmic scales.
Real-World Applications of Logarithms
Logarithms appear throughout science and everyday life. The pH scale measures acidity using pH = -log₁₀[H⁺], where each unit represents a tenfold change in hydrogen ion concentration. The Richter scale for earthquakes is logarithmic: a magnitude 6 earthquake releases about 31.6 times more energy than magnitude 5. Sound intensity is measured in decibels using dB = 10 × log₁₀(I/I₀). In finance, logarithms help calculate compound interest and continuous growth. Information theory uses logarithms to measure entropy and data compression efficiency. Even biological phenomena like human perception of brightness and loudness follow logarithmic scales. For financial calculations, our Compound Interest Calculator can help with growth projections.
Solving Logarithmic Equations
To solve logarithmic equations, use the definition of logarithms and their properties. For equations like log_b(x) = y, convert to exponential form: x = b^y. For equations with multiple logarithms, use properties to combine them into a single logarithm, then solve. Always check solutions by substituting back, as logarithms are only defined for positive arguments. When solving exponential equations like 2^x = 10, take the logarithm of both sides: x = log₂(10) = ln(10)/ln(2). Remember that logarithmic and exponential functions are inverses, so they 'undo' each other. For solving polynomial equations, try our Quadratic Formula Calculator.
Pro Tips
- 💡Remember that log_b(b) = 1 for any valid base - this is a quick mental check for your calculations.
- 💡When multiplying numbers with many digits, add their logarithms and find the antilog - this historical technique still aids understanding.
- 💡The natural log of e is exactly 1, and ln(1) is exactly 0 - memorize these anchor points.
- 💡To estimate log₁₀ of a number, count digits minus one: log₁₀(1000) ≈ 3 (four digits, minus one).
- 💡Use logarithms to compare very large or very small numbers more easily - the difference in logs shows orders of magnitude.
- 💡In the change of base formula, you can use any base for conversion - choose whichever is most convenient.
- 💡Logarithmic scales compress large ranges: pH 1 to 14 actually spans 10^14 fold change in H⁺ concentration.
- 💡When solving log equations, always verify solutions work in the original equation since log is only defined for positive numbers.
- 💡The binary logarithm (log₂) is essential in computer science - it tells you how many bits are needed to represent a number.
- 💡Practice converting between logarithmic and exponential forms: log_b(x) = y is equivalent to b^y = x.
- 💡For quick mental math, remember that log₁₀(2) ≈ 0.301, so doubling a number adds about 0.301 to its common log.
- 💡Logarithms turn multiplication into addition and division into subtraction - this is why slide rules worked.
Frequently Asked Questions
A logarithm tells you what exponent (power) you need to raise a base number to in order to get another number. For example, log₁₀(100) = 2 because 10² = 100. Think of it as asking: '10 to what power equals 100?' The answer is 2.