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

115

Calculation Results

Enter a positive number to calculate negative logarithm

Common Applications

pH

pH Calculations

pH = -log₁₀([H⁺])

Measures acidity/basicity

dB

Decibels

Sound intensity scales

Audio engineering

pK

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.