Negative Log Calculator
Calculate negative logarithms with any base - commonly used in pH calculations and data analysis
Calculate Negative Logarithm
Must be a positive number (x > 0)
Base must be positive and not equal to 1
Calculation Results
Common Applications
pH Calculations
pH = -log₁₀([H⁺])
Measures acidity/basicity
Decibels
Sound intensity scales
Audio engineering
pKa/pKb Values
Acid/base strength
Chemical equilibrium
Quick Reference
Common Bases
- Base 10: Common logarithm
- Base e: Natural logarithm
- Base 2: Binary logarithm
Key Properties
- -log(1) = 0
- -log(x) = log(1/x)
- -log(x·y) = -log(x) - log(y)
Example: pH Calculation
Problem
Find the pH of a solution with [H⁺] = 0.001 M
Solution
pH = -log₁₀([H⁺])
pH = -log₁₀(0.001)
pH = -log₁₀(10⁻³)
pH = 3
Understanding Negative Logarithms
What is a Negative Logarithm?
A negative logarithm is simply the negative of a regular logarithm. If log_a(b) = n, then -log_a(b) = -n. It's not the logarithm of a negative number (which is undefined for real numbers), but rather the negative of a logarithm.
Mathematical Definition
-log_a(b) = n
means a^(-n) = b, or a^n = 1/b
Key Properties
- •-log_a(1) = 0 (since log_a(1) = 0)
- •-log_a(x) = log_a(1/x)
- •-log_a(x·y) = -log_a(x) - log_a(y)
- •For x > 1: -log_a(x) < 0
- •For 0 < x < 1: -log_a(x) > 0
Real-World Applications
Chemistry: pH Scale
pH = -log₁₀([H⁺])
Measures hydrogen ion concentration in solutions. Lower pH = more acidic.
Physics: Decibels
dB = -20 log₁₀(V₁/V₂)
Used in acoustics and electronics to measure ratios on a logarithmic scale.
Data Science: Information Theory
Entropy = -Σ p(x) log₂(p(x))
Measures information content and uncertainty in data.
Important Note
Negative logarithm (-log_a(x)) is different from logarithm of negative number (log_a(-x)). The latter is undefined for real numbers but can be computed using complex numbers.