Logarithm Calculator
Calculate Log base 10, Natural Log (ln), or any custom base.
Log₁₀(x)
Log Base 10
Natural Log (ln)
Logarithmic Curve (y = logbx)
Log Rules Cheat Sheet
-
Product Rule
log(xy) = log(x) + log(y) -
Quotient Rule
log(x/y) = log(x) - log(y) -
Power Rule
log(x^p) = p × log(x)
What is a Logarithm?
A logarithm answers the question: "How many times must we multiply a specific number (the base) to get another number?" It is the inverse operation to exponentiation.
[Image of logarithm formula]If by = x, then logb(x) = y.
- Example: Since 10 × 10 = 100 (or 102 = 100), then log10(100) = 2.
- Example: Since 2 × 2 × 2 = 8 (or 23 = 8), then log2(8) = 3.
Common Types of Logarithms
1. Common Logarithm (log)
Uses Base 10. If you see "log(x)" written without a base, it usually means base 10. It is widely used in science and engineering (e.g., pH scale, Richter scale for earthquakes, Decibels for sound).
2. Natural Logarithm (ln)
Uses Base e (Euler's number ≈ 2.718). It is written as "ln(x)". It is fundamental in calculus, physics, and calculating compound interest/growth.
3. Binary Logarithm (log₂)
Uses Base 2. This is the language of computers (bits and bytes). It is crucial in computer science and information theory.
Why can't we find the Log of a negative number?
You cannot find the logarithm of a negative number or zero (for real numbers). Why?
Look at the formula: by = x.
If the base `b` is positive (like 10), then `10` to the power of *any* number (positive or negative) will *always* be positive.
102 = 100. 10-2 = 1/100 = 0.01.
You can never get a negative result or zero from a positive base. The graph above shows this clearly—the line never touches the left side (negative x).