Logarithm Calculator

How to Use This Logarithm Calculator

1. Enter the number: Input the positive number (x) you want to find the logarithm of. Logarithms are only defined for positive numbers.
2. Select or enter the base: Choose a common base using the quick buttons (log₁₀, ln, log₂) or enter any positive base greater than 0 and not equal to 1.
3. Calculate: Click "Calculate Logarithm" to compute log_b(x) - the logarithm of x with base b.
4. View steps: See detailed explanations including the mathematical notation, exponential form, and verification.

What is a Logarithm?

A logarithm answers the question: "To what power must we raise the base to get this number?" In other words, if b^y = x, then log_b(x) = y. For example, log₁₀(100) = 2 because 10² = 100.

Logarithms are the inverse operation of exponentiation, just as subtraction is the inverse of addition and division is the inverse of multiplication. They transform multiplication into addition and exponentiation into multiplication, making complex calculations more manageable.

Common Types of Logarithms

Common logarithm (log₁₀): Base 10 logarithm, written as log(x) or log₁₀(x). Used in pH calculations, decibel measurements, and the Richter scale. Example: log₁₀(1000) = 3.

Natural logarithm (ln): Base e (≈2.71828) logarithm, written as ln(x) or log_e(x). Fundamental in calculus, exponential growth/decay, and compound interest. Example: ln(e²) = 2.

Binary logarithm (log₂): Base 2 logarithm, written as log₂(x). Essential in computer science for analyzing algorithms, data structures, and information theory. Example: log₂(8) = 3.

Logarithm Properties and Rules

Product rule: log_b(xy) = log_b(x) + log_b(y). The logarithm of a product equals the sum of logarithms. Example: log(100) = log(10×10) = log(10) + log(10) = 1 + 1 = 2.

Quotient rule: log_b(x/y) = log_b(x) - log_b(y). The logarithm of a quotient equals the difference of logarithms. Example: log(10/2) = log(10) - log(2).

Power rule: log_b(x^n) = n × log_b(x). The logarithm of a power equals the exponent times the logarithm. Example: log(10³) = 3 × log(10) = 3.

Change of base formula: log_b(x) = log_a(x) / log_a(b). Convert between bases using this formula. Example: log₂(8) = log₁₀(8) / log₁₀(2).

Special Logarithm Values

Logarithm of 1: log_b(1) = 0 for any base b. Any number raised to the power 0 equals 1, so log(1) is always 0.

Logarithm of the base: log_b(b) = 1. The base raised to the power 1 equals itself, so log₁₀(10) = 1, ln(e) = 1, log₂(2) = 1.

Logarithm of base to a power: log_b(b^n) = n. For example, log₁₀(10⁵) = 5, ln(e³) = 3.

Undefined values: Logarithms of zero and negative numbers are undefined in real numbers. log_b(0) and log_b(negative) require complex numbers.

Real-World Applications

Logarithms appear throughout science and everyday life. In chemistry, pH measures acidity as -log[H⁺], where small changes in pH represent large changes in acidity. In acoustics, decibels measure sound intensity logarithmically: dB = 10×log₁₀(I/I₀). Earthquakes use the Richter scale where each whole number represents a 10-fold increase in amplitude. In biology, logarithmic scales describe population growth, enzyme kinetics, and evolutionary timescales. Finance uses natural logarithms for continuous compound interest and stock returns. Computer science employs log₂ extensively in algorithm analysis, where O(log n) algorithms are highly efficient.

Logarithms are the inverse of exponents - use the Exponent Calculator for power calculations. The Scientific Calculator includes log functions along with other advanced operations.