Logarithm Calculator
Compute logarithms in different forms: base-b log, natural log (ln), common log (log10), and antilog.
What is a logarithm?
A logarithm answers a simple but powerful question: “What exponent gives this number?” If we write by = x, then y = logb(x). In plain language, the logarithm is the inverse operation of exponentiation.
Quick examples
- log10(100) = 2 because 102 = 100
- log2(8) = 3 because 23 = 8
- ln(e) = 1 because e1 = e
How to use this calculator
This page supports four common logarithm-related calculations:
- log_b(x): General logarithm with custom base.
- ln(x): Natural log, base e.
- log10(x): Common logarithm, base 10.
- Antilog b^y: Inverse of log; computes exponentiation directly.
Input rules you should know
- For logarithms, the value x must be positive (x > 0).
- The base b must be positive and not equal to 1.
- Decimal places can be set from 0 to 12 for cleaner output.
Why logarithms matter in real life
Logarithms appear whenever values grow quickly or span huge ranges. They let us compress large data and reason about exponential behavior in a practical way.
Common applications
- Finance: Compound growth and continuous compounding models.
- Science: pH scale, earthquake magnitude, sound intensity (decibels).
- Computer science: Algorithm complexity and binary search patterns.
- Statistics: Log transforms for skewed data distributions.
Change-of-base formula
Most calculators and programming languages use natural logarithms internally. So custom-base logs are commonly computed with:
That is exactly how this tool evaluates logb(x) behind the scenes.
Worked mini examples
Example 1: log3(81)
We seek y such that 3y = 81. Since 34 = 81, the result is 4.
Example 2: ln(20)
Natural log of 20 is approximately 2.995732 (depending on rounding).
Example 3: Antilog with base 10 and exponent 2.5
102.5 = 316.227766... This is useful when converting from log-scale values back to standard scale.
Common mistakes to avoid
- Using x = 0 or x < 0 in logarithms.
- Choosing base 1 (invalid, because 1 raised to any power is still 1).
- Confusing ln(x) and log10(x) in formulas.
- Rounding too early in multi-step calculations.
Final thoughts
A good logarithm calculator should be fast, transparent, and strict with math rules. Use this tool to check homework, validate formulas, or build intuition for exponential relationships. If you work with growth rates, data scales, or scientific measurements, logarithms are not optional—they are essential.