Log Calculator
Calculate logarithms with any base.
Logarithm Input
Must be greater than 0
Must be greater than 0 and not equal to 1
Expression:
log_10(100)
Logarithm Results
log_10(100)
2.0000
Natural Log: ln(100)
4.6052
Common Log: log₁₀(100)
2.0000
Binary Log: log₂(100)
6.6439
Verification:
10^2.0000 ≈ 100.00
About This Calculator
The Log Calculator helps you calculate logarithms with any base. Logarithms are the inverse operation of exponentiation and are widely used in mathematics, science, and engineering.
The calculator supports natural logarithm (ln, base e), common logarithm (log₁₀, base 10), binary logarithm (log₂, base 2), and logarithms with any custom base. Logarithms are useful for solving exponential equations and working with very large or very small numbers.
Formula & Calculation Method
Logarithm Formulas:
Logarithm Definition:
log_b(x) = y means b^y = x
Change of Base Formula:
log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b)
Common Logarithms:
ln(x) = log_e(x) - natural logarithm (base e ≈ 2.718)
log₁₀(x) - common logarithm (base 10)
log₂(x) - binary logarithm (base 2)
Properties:
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)
Logarithms are the inverse of exponentials. If b^y = x, then log_b(x) = y. The change of base formula allows calculating logarithms with any base using natural or common logarithms.
How to Use This Calculator
- Enter Value: Input the number you want to take the logarithm of (must be positive).
- Enter Base: Input the base of the logarithm (must be positive and not equal to 1).
- View Results: See the logarithm with your base, plus natural log, common log, and binary log.
- Special Bases: Use base 10 for common log, base e (2.718...) for natural log, base 2 for binary log.
Frequently Asked Questions
What is a logarithm?
A logarithm is the inverse operation of exponentiation. If b^y = x, then log_b(x) = y. Logarithms answer the question: "To what power must I raise the base to get this value?"
What are natural and common logarithms?
Natural logarithm (ln) uses base e (approximately 2.718), commonly used in calculus and science. Common logarithm (log₁₀) uses base 10, often used in engineering and for scientific notation.
Why can't the base be 1?
If the base is 1, then 1 raised to any power equals 1, so log_1(x) would be undefined for x ≠ 1, and any value for x = 1. Therefore, base 1 is not valid for logarithms.
How are logarithms used in real life?
Logarithms are used in many applications: measuring sound (decibels), measuring earthquakes (Richter scale), calculating compound interest, analyzing exponential growth/decay, and solving exponential equations.
NumCalculators Editorial Team
Multi-disciplinary Expert Team
The NumCalculators Editorial Team consists of subject matter experts across various fields who review and verify the accuracy of our calculators. Each calculator undergoes rigorous testing and validation before publication.