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Logarithm Calculator
Calculate logarithms with different bases. Essential for advanced mathematics, science, and engineering applications.
Enter Values
Provide the number and base for logarithm calculation
The number you want to find the logarithm of (must be positive)
Choose the type of logarithm to calculate
The base of the logarithm (must be positive and not equal to 1)
Logarithm Results
Calculated logarithm values
Enter your values and click "Calculate Logarithm" to see your results
Common Logarithm Values
Reference values for common bases and numbers
Base 10 (Common)
log₁₀(1)0
log₁₀(10)1
log₁₀(100)2
log₁₀(1000)3
Base e (Natural)
ln(1)0
ln(e)1
ln(e²)2
ln(10)2.3026
Base 2 (Binary)
log₂(1)0
log₂(2)1
log₂(4)2
log₂(8)3
About Logarithms
What are Logarithms?
Logarithms are the inverse operation to exponentiation. The logarithm of a number x to a given base b is the exponent to which the base must be raised to produce the number x. In other words, if bʸ = x, then y = log_b(x).
Types of Logarithms
Common Types:
- • Natural Logarithm (ln): Base e ≈ 2.71828, widely used in calculus and natural sciences
- • Common Logarithm (log₁₀): Base 10, commonly used in engineering and decimal systems
- • Binary Logarithm (log₂): Base 2, essential in computer science and information theory
- • Custom Base: Any positive base b ≠ 1, used in various mathematical contexts
Logarithm Properties
Basic Properties
- • log_b(1) = 0
- • log_b(b) = 1
- • log_b(bˣ) = x
- • b^(log_b(x)) = x
Arithmetic Properties
- • log_b(xy) = log_b(x) + log_b(y)
- • log_b(x/y) = log_b(x) - log_b(y)
- • log_b(xʳ) = r × log_b(x)
- • log_b(x) = log_c(x) / log_c(b)
Applications of Logarithms
- Mathematics: Solving exponential equations, calculus, complex analysis
- Science: pH calculations, radioactive decay, population growth, sound intensity (decibels)
- Engineering: Signal processing, control systems, earthquake magnitude (Richter scale)
- Computer Science: Algorithm complexity, information theory, data compression
- Finance: Compound interest, investment growth, economic modeling
Domain and Range
Important Constraints:
- • Domain: x > 0 (logarithms are only defined for positive numbers)
- • Base: b > 0 and b ≠ 1 (base must be positive and not equal to 1)
- • Range: All real numbers (-∞ to +∞)
- • Special Cases: log_b(1) = 0 for any valid base b
Tips for Using This Calculator
- Enter positive numbers only - logarithms are undefined for zero and negative numbers
- For custom base logarithms, ensure the base is positive and not equal to 1
- Natural logarithms use base e ≈ 2.71828, the mathematical constant
- Common logarithms use base 10, useful for decimal-based calculations
- The exponential form shows the inverse relationship: if y = log_b(x), then bʸ = x