Log Calculator (Logarithm)

Calculate logarithms with any base instantly. Solve for x, base, or result with step-by-step solutions and AI-powered insights.

🧮 Log Calculator (Logarithm)

What is a Logarithm?

A logarithm is the inverse operation of exponentiation. It answers the question: 'To what power must we raise the base to get a certain number?' For example, log₁₀(100) = 2 because 10² = 100.

Logarithms are fundamental in mathematics, science, and engineering. They help us work with exponential growth and decay, solve complex equations, and understand phenomena that span multiple orders of magnitude.

Our Log Calculator supports common logarithms (base 10), natural logarithms (base e), binary logarithms (base 2), and custom bases. You can solve for any of the three variables: the value (x), the base (b), or the result (y).

How to Use the Log Calculator

Enter Known Values: Fill in at least two of the three fields: Value (x), Base (b), or Result (y). Leave the field you want to calculate blank.
Choose a Base: Use the preset buttons for common bases (10, e, 2) or enter a custom base. The base must be positive and not equal to 1.
Click Calculate: The calculator will compute the missing value and display the result with a detailed formula and explanation.
Review Results: See the answer, the mathematical formula used, and an explanation of what the result means in context.

Latest Insights on Logarithms

Logarithms compute the exponent in exponential equations like log_b(x) = y where b^y = x, commonly using base 10 (log), base e (ln), or custom bases.
They are essential in science, engineering, finance, chemistry (pH calculations), physics (decibels), statistics, and computer science for handling exponential growth/decay and data analysis.
Modern calculators like Casio fx-991EX and TI-84 offer direct custom base functions with accuracy up to 10+ decimals, supporting change-of-base formulas: log_b(x) = log_a(x) / log_a(b).
Best practices include verifying input > 0 and base > 0 ≠ 1, using inverse functions (10^x, e^x) to check results, and applying log rules like product rule (log(xy) = log x + log y).
Logarithms model exponential processes in health/fitness contexts like bacterial growth in nutrition or decay in pharmacokinetics, requiring high precision for scientific applications.

Understanding Logarithms in Detail

Common Logarithm Bases

Base 10 (Common Logarithm): Written as log(x) or log₁₀(x). Used extensively in engineering, science, and the Richter scale for earthquakes.
Base e (Natural Logarithm): Written as ln(x) or logₑ(x), where e ≈ 2.71828. Essential in calculus, continuous growth models, and physics.
Base 2 (Binary Logarithm): Written as log₂(x). Fundamental in computer science, information theory, and binary systems.

Logarithm Rules and Properties

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 Formula: log_b(x) = log_a(x) / log_a(b)

Real-World Applications

pH Calculations: pH = -log₁₀[H⁺] measures acidity in chemistry
Decibel Scale: Sound intensity measured as dB = 10 · log₁₀(I/I₀)
Richter Scale: Earthquake magnitude uses logarithmic scale
Finance: Compound interest and exponential growth calculations
Data Compression: Information theory and entropy calculations

Frequently Asked Questions

What is the difference between log and ln?

log typically refers to the common logarithm (base 10), while ln refers to the natural logarithm (base e ≈ 2.71828). Both are logarithms, just with different bases.

Can I calculate logarithms with negative numbers?

No, logarithms are only defined for positive real numbers. The value (x) must be greater than 0, and the base must be positive and not equal to 1.

How do I convert between different logarithm bases?

Use the change-of-base formula: log_b(x) = log_a(x) / log_a(b). For example, to convert log₂(8) to base 10: log₂(8) = log₁₀(8) / log₁₀(2).

What does it mean when log_b(x) = y?

It means that b raised to the power of y equals x. In other words, b^y = x. For example, log₁₀(100) = 2 means 10² = 100.

Why can't the base be 1?

If the base were 1, then 1 raised to any power would always equal 1, making the logarithm undefined or meaningless. The base must be positive and not equal to 1.