Logarithm Calculator
Calculate logarithms with any base including natural log (ln) and common log (log₁₀). Get step-by-step solutions and mathematical insights.
Logarithm Input
Enter a positive number (x > 0)
Logarithm Results
Enter a positive number to calculate its logarithm
How to Use the Logarithm Calculator
Enter Your Number
Type the positive number for which you want to find the logarithm. This is the argument of the logarithm function. Remember that logarithms are only defined for positive numbers.
Select Logarithm Type
Choose between Common Logarithm (base 10), Natural Logarithm (base e), or Custom Base. Each type has different applications in mathematics and science.
Set Custom Base (if needed)
For custom base logarithms, enter the base value. The base must be positive and not equal to 1. Common custom bases include 2 (binary), 3, 5, etc.
View Results and Steps
The calculator shows the logarithm result, step-by-step solution, antilog verification, and visualization. Use these to understand the mathematical process.
Logarithm Tips & Properties
Logarithm Definition: logₑ(x) = y means b^y = x. The logarithm answers 'what power must b be raised to get x?'
Domain Restriction: Logarithms are only defined for positive numbers. You cannot take the log of zero or negative numbers.
Common Logarithm: log₁₀(x) is called the common logarithm, often written as log(x). Used in scientific calculations and pH measurements.
Natural Logarithm: ln(x) = logₑ(x) where e ≈ 2.71828. Widely used in mathematics, physics, and engineering.
Change of Base Formula: log₍ₐ₎(x) = ln(x) / ln(a) = log(x) / log(a). Useful for calculating logs with any base.
Logarithm Properties: log(a×b) = log(a) + log(b), log(a/b) = log(a) - log(b), log(a^n) = n×log(a)
Inverse Relationship: Logarithms and exponentials are inverse functions. If y = log₍ₐ₎(x), then x = a^y (antilog).
Special Values: log₍ₐ₎(1) = 0 for any base a > 0, log₍ₐ₎(a) = 1, log₍ₐ₎(a^n) = n