Logarithm Calculator
What is a Logarithm?
A logarithm is the inverse operation to exponentiation. It answers the question: "To what power must a given base be raised to produce a given number?"
If b^y = x, then log_b(x) = y, where b is the base of the logarithm.
log_b(x) = y means b^y = x
Common Types of Logarithms
Natural Logarithm (ln)
The natural logarithm has base e (approximately 2.71828), which is an irrational number that appears frequently in mathematics and natural sciences.
ln(x) = log_e(x)
Common Logarithm (log)
The common logarithm has base 10. It is often written simply as "log" without explicitly stating the base.
log(x) = log_10(x)
Binary Logarithm (log₂)
The binary logarithm has base 2. It is commonly used in computer science and information theory.
log₂(x)
Custom Base Logarithm
Logarithms can have any positive base (except 1). Different bases are useful in different contexts.
log_b(x) for any b > 0, b ≠ 1
Logarithm Properties
Product Rule
log_b(x × y) = log_b(x) + log_b(y)
The logarithm of a product equals the sum of the logarithms of the factors.
Quotient Rule
log_b(x / y) = log_b(x) - log_b(y)
The logarithm of a quotient equals the logarithm of the numerator minus the logarithm of the denominator.
Power Rule
log_b(x^n) = n × log_b(x)
The logarithm of a power equals the exponent times the logarithm of the base.
Change of Base Formula
log_b(x) = log_c(x) / log_c(b)
This formula allows you to calculate logarithms with any base using logarithms with another base.
Applications of Logarithms
Logarithms have numerous applications in various fields:
- Science and Engineering: Measuring quantities that vary over a wide range (pH, decibels, Richter scale)
- Finance: Calculating compound interest and analyzing financial growth
- Computer Science: Analyzing algorithm efficiency and data compression
- Statistics: Working with normal distributions and performing data transformations
- Information Theory: Measuring information content and entropy
Logarithm Rules and Examples
Basic Logarithm Values
- log_b(1) = 0 (because b^0 = 1)
- log_b(b) = 1 (because b^1 = b)
- log_b(b^n) = n
Example Calculations
- log_10(100) = 2 (because 10^2 = 100)
- log_2(8) = 3 (because 2^3 = 8)
- ln(e) = 1 (because e^1 = e)
- log_10(0.01) = -2 (because 10^(-2) = 0.01)