Negative Log — -log for pH, pKa, pOH
pH = -log₁₀([H⁺]). pKa, pOH, information content. Step-by-step solutions and pH scale visualization.
Negative Logarithm — pH Scale & Beyond
Compute -log_b(x) for pH, pKa, pOH, information theory, and signal processing. From chemistry to Shannon entropy.
🧪 Quick Examples — Click to Load
Enter Values
Compute — value x must be positive, base b > 0 and ≠ 1.
⚠️For educational and informational purposes only. Verify with a qualified professional.
📋 Key Takeaways
- • -log_b(x) converts small positive numbers (0<x<1) into positive, interpretable values
- • pH = -log₁₀[H⁺] — each pH unit is a 10× change in acidity
- • pKa and pKb measure acid/base strength in pharmacology
- • Information content I(p) = -log₂(p) bits — rarer events carry more information
- • Decibels (dB) use logarithmic ratios; -log appears in signal attenuation
💡 Did You Know?
📖 How It Works
The negative logarithm flips the sign of log_b(x). For 0<x<1, log(x) is negative, so -log(x) is positive.
pH Chemistry
pH = -log₁₀[H⁺]. [H⁺] = 10⁻⁷ → pH = 7 (neutral). Each unit = 10× change.
Information Theory
I(p) = -log₂(p) bits. Probability ½ → 1 bit; ¼ → 2 bits.
🎯 Expert Tips
💡 pH Quick Check
pH 7 = neutral. pH < 7 acidic, pH > 7 basic. [H⁺]=10⁻³ → pH=3 (strong acid).
💡 pKa in Drug Design
Acids with pKa < 3 are mostly ionized at stomach pH. Bases with pKa > 8 absorb in intestine.
💡 Information Content
-log₂(p) gives bits. Surprisal of a 1-in-1024 event = 10 bits. Used in entropy and compression.
💡 Decibel & Attenuation
dB = 10·log₁₀(P₂/P₁). Negative dB = loss. -3 dB ≈ half power.
⚖️ Comparison Table
| Application | Formula | Base |
|---|---|---|
| pH | -log₁₀[H⁺] | 10 |
| pOH | -log₁₀[OH⁻] | 10 |
| pKa | -log₁₀(K_a) | 10 |
| Information (bits) | -log₂(p) | 2 |
| Decay (natural) | -ln(λ) | e |
❓ Frequently Asked Questions
Why use negative log for pH?
Hydrogen ion concentrations are tiny (e.g., 10⁻⁷ M). Taking -log₁₀ converts them to manageable positive numbers (pH 7). Each pH unit is a 10× change in [H⁺].
What is pKa vs pH?
pH measures acidity of a solution. pKa measures the strength of an acid (its tendency to donate H⁺). pKa = -log₁₀(K_a), where K_a is the acid dissociation constant.
When is -log negative?
When x > 1. For example, -log₁₀(10) = -1. Most applications (pH, pKa, information) use 0 < x < 1, giving positive -log.
What is information content in bits?
I(p) = -log₂(p). A fair coin (p=½) gives 1 bit. A 1-in-8 event gives 3 bits. It quantifies "surprise" or information gained when the event occurs.
How does -log relate to decibels?
dB = 10·log₁₀(P₂/P₁). When P₂ < P₁ (attenuation), the ratio < 1, so log is negative. The negative indicates loss. -3 dB ≈ half power.
Can I use -log for any base?
Yes. Base 10 for pH/pKa, base 2 for information (bits), base e for natural decay. The formula -log_b(x) = -ln(x)/ln(b) works for any valid base.
📊 Key Constants
📚 Reference Sources
⚠️ Note: This calculator is for educational purposes. For clinical or laboratory pH/pKa measurements, use calibrated instruments. Results use IEEE 754 double-precision; expect ~15 significant digits.