Log Calculator
What are the key logarithm rules and identities?
| Rule | Formula | Example |
|---|---|---|
| Product rule | log(a × b) = log(a) + log(b) | log(6) = log(2) + log(3) |
| Quotient rule | log(a ÷ b) = log(a) − log(b) | log(5) = log(10) − log(2) |
| Power rule | log(aⁿ) = n × log(a) | log(8) = 3 × log(2) |
| Change of base | log_b(x) = log(x) ÷ log(b) | log₂(10) = 1 ÷ log(2) = 3.322 |
| Inverse (antilog) | if log(x) = y, then x = 10^y | log(x) = 2 → x = 100 |
| log₁₀(1) = 0 | 10⁰ = 1 | Applies to any base |
| log(base) = 1 | log₁₀(10) = 1; ln(e) = 1 | Any base to itself = 1 |
Common logarithm values reference table
| Number | log₁₀ | ln (natural log) | log₂ |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 0.3010 | 0.6931 | 1 |
| e (2.718...) | 0.4343 | 1 | 1.4427 |
| 10 | 1 | 2.3026 | 3.3219 |
| 100 | 2 | 4.6052 | 6.6439 |
| 1,000 | 3 | 6.9078 | 9.9658 |
| 1,000,000 | 6 | 13.8155 | 19.9316 |
Where are logarithms used in real life?
Finance: Compound interest calculations, CAGR, Rule of 72 (years = 72 ÷ rate is an approximation of ln(2) ÷ rate). Science: pH scale (pH = −log[H⁺]), earthquake magnitude (Richter scale), sound decibels (dB = 10 × log(P/P₀)). Computing: Binary logarithm (log₂) for algorithm complexity (O(log n) search). Music: Musical intervals in cents = 1,200 × log₂(f₂/f₁). Related: use the Exponent Calculator to go further, or the Scientific Calculator for a different angle.
Frequently asked questions
What is the difference between log and ln?
log (or log₁₀) is the common logarithm with base 10. ln is the natural logarithm with base e (≈2.71828). log₁₀(100) = 2. ln(100) = 4.6052. In mathematics and science, 'log' without a base often means natural log. In engineering and common use, 'log' usually means log₁₀. Always check the context.
What is an antilog?
Antilog is the inverse of logarithm. If log₁₀(x) = 2, then antilog(2) = 10² = 100. If ln(x) = 3, then e³ = 20.086. Antilog is simply raising the base to the power of the logarithm value: antilog_b(y) = b^y.
How do you calculate log of a fraction?
log(a/b) = log(a) − log(b). log(0.1) = log(1/10) = log(1) − log(10) = 0 − 1 = −1. log(0.01) = −2. Negative logarithms always correspond to numbers between 0 and 1. log(0) is undefined (there is no power that makes 10 equal to 0).
What base does a scientific calculator use for 'log'?
All scientific calculators use base 10 for the 'LOG' button and base e for the 'LN' button. To calculate log base 2 of 64 on a calculator: log(64) ÷ log(2) = 1.806 ÷ 0.301 = 6. Verify: 2⁶ = 64 ✓
How is the Richter scale calculated using logarithms?
Each 1-point increase on the Richter scale represents 10× more ground motion amplitude. Formula: M = log₁₀(A) − log₁₀(A₀). A magnitude 6.0 earthquake releases 31.6× more energy than magnitude 5.0. A magnitude 7.0 releases 1,000× more energy than 5.0 — the logarithmic scale compresses enormous ranges into manageable numbers.
Sources & References
Sources: NIST Digital Library of Mathematical Functions — Chapter 4 (Elementary Functions); Abramowitz & Stegun Handbook of Mathematical Functions; ISO 80000-2:2019 (Mathematical signs and symbols).
Real-world logarithm applications
pH scale: chemistry and biology
pH = −log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration in mol/L. Each pH unit is a 10-fold change in acidity. pH 7 = neutral (pure water). Below 7 = acidic. Above 7 = alkaline.
| Substance | [H⁺] mol/L | pH | Category |
|---|---|---|---|
| Battery acid | 1 | 0 | Extremely acidic |
| Lemon juice | 0.01 | 2 | Strongly acidic |
| Vinegar | 0.001 | 3 | Acidic |
| Black coffee | 10⁻⁵ | 5 | Mildly acidic |
| Pure water | 10⁻⁷ | 7 | Neutral |
| Blood | 10⁻⁷·⁴ | 7.4 | Slightly alkaline |
| Baking soda | 10⁻⁹ | 9 | Alkaline |
| Bleach | 10⁻¹² | 12 | Strongly alkaline |
Richter scale: earthquake magnitude
M = log₁₀(A/A₀), where A is amplitude and A₀ is a reference. Each 1-point increase = 10× more ground shaking amplitude = about 31.6× more energy released. Magnitude 5.0 vs 3.0: 10² = 100× more shaking, 31.6² ≈ 1,000× more energy. The 2004 Indian Ocean earthquake (M 9.1) released roughly 1,000× more energy than the 2001 Gujarat earthquake (M 7.7).
Sound decibels: hearing and noise levels
dB = 10 × log₁₀(P/P₀), where P₀ = 10⁻¹² W/m² (threshold of hearing). Each 10 dB increase = 10× more sound power.
| Sound | dB Level | Times louder than threshold |
|---|---|---|
| Rustling leaves | 20 dB | 100× |
| Normal conversation | 60 dB | 1,000,000× |
| Heavy traffic | 80 dB | 100,000,000× |
| Jet engine (30m away) | 130 dB | 10,000,000,000,000× |
| Threshold of pain | 140 dB | 100 trillion× |
Hearing damage begins at sustained exposure above 85 dB. Without logarithms, these enormous ratios would be impossible to express on a practical scale. Related: Scientific Calculator · Exponent Calculator.
Log tables: how calculations were done before calculators
Before electronic calculators (pre-1970s), scientists and engineers used printed log tables to perform multiplication, division, and exponentiation. The method: to multiply 347 × 829, look up log(347) = 2.5403 and log(829) = 2.9186. Add: 2.5403 + 2.9186 = 5.4589. Find the antilog: 10^5.4589 = 287,963 ≈ 287,600. This reduced multiplication to addition — far faster by hand. John Napier published the first log tables in 1614. Slide rules used the same principle physically. The scientific calculator (1972) made log tables obsolete, but understanding logarithms remains essential for interpreting exponential data in any field.
Logarithm memorisation tricks for exams
Key values to memorise: log₁₀(2) = 0.3010. log₁₀(3) = 0.4771. From these, derive others: log(4) = log(2²) = 2×0.3010 = 0.6020. log(5) = log(10/2) = 1−0.3010 = 0.6990. log(6) = log(2)+log(3) = 0.3010+0.4771 = 0.7781. log(8) = 3×log(2) = 0.9030. log(9) = 2×log(3) = 0.9542. Natural log shortcuts: ln(2) ≈ 0.693. ln(10) ≈ 2.303. Change of base: log_b(x) = ln(x)/ln(b) = log(x)/log(b). These four memorised values generate most common logarithm results needed in exam questions. Related: Scientific Calculator.
Half-life calculations using natural logarithm
Radioactive decay and drug elimination use the natural logarithm. Half-life formula: t½ = ln(2) ÷ λ, where λ is the decay constant. Carbon-14 has λ = 0.000121/year, so t½ = 0.693 ÷ 0.000121 = 5,730 years. For drug elimination: a medicine with a 4-hour half-life: after 4 hours, 50% remains. After 8 hours, 25%. After 24 hours, 1.56% (6 half-lives). Formula: remaining = initial × (0.5)^(time/half-life) = initial × e^(−λ×time). These formulas appear in biology, pharmacology, archaeology (carbon dating), and nuclear engineering. The key is that ln(2) ≈ 0.693 — the natural log of 2 — is the universal constant connecting decay rate to half-life.
Logarithms in real life: pH, earthquakes, and sound
Logarithms are used whenever a quantity spans many orders of magnitude. Three classic applications:
pH and chemistry
pH = −log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration in mol/L. A pH difference of 1 = a 10× difference in acidity. pH 2 is 10× more acidic than pH 3, and 100× more acidic than pH 4.
| Substance | [H⁺] mol/L | pH | Category |
|---|---|---|---|
| Battery acid | 1 × 10⁰ | 0 | Strongly acidic |
| Stomach acid | 1 × 10⁻² | 2 | Acidic |
| Lemon juice | 1 × 10⁻²·⁵ | 2.5 | Acidic |
| Coffee | 1 × 10⁻⁵ | 5 | Weakly acidic |
| Pure water | 1 × 10⁻⁷ | 7 | Neutral |
| Baking soda | 1 × 10⁻⁸·³ | 8.3 | Weakly alkaline |
| Bleach | 1 × 10⁻¹² | 12 | Strongly alkaline |
Logarithms in seismology: the Richter scale explained
Magnitude = log₁₀(amplitude of seismic waves). Each integer step = 10× more ground motion and ≈31.6× more energy released. A magnitude 7.0 earthquake releases 31.6× more energy than a 6.0, and 1,000× more energy than a 5.0. The 2001 Gujarat earthquake (M 7.7) released approximately 31.6³ ≈ 31,600× more energy than a M 4.7 minor tremor.
Decibels: sound intensity
dB = 10 × log₁₀(I/I₀), where I₀ = 10⁻¹² W/m² (threshold of human hearing). A 10 dB increase = 10× more sound intensity, but only about 2× louder to human perception.
| Sound | dB Level | Intensity ratio vs hearing threshold |
|---|---|---|
| Whisper | 30 dB | 1,000× |
| Normal conversation | 60 dB | 1,000,000× |
| Busy traffic | 85 dB | 316,000,000× |
| Rock concert | 110 dB | 100,000,000,000× |
| Jet engine at 30m | 140 dB | 100,000,000,000,000× |
ln vs log: which fields use which base?
| Field | Preferred Base | Reason |
|---|---|---|
| Pure mathematics | ln (base e) | Natural derivative: d/dx ln(x) = 1/x (no constants) |
| Engineering, physics | log₁₀ | Easy mental conversion to powers of 10 |
| Computer science | log₂ | Binary systems; number of bits needed = ⌈log₂(n)⌉ |
| Finance / economics | ln | Continuous compounding: A = Pe^(rt) uses e |
| Chemistry (pH, pKa) | log₁₀ | pH = −log[H⁺] |
| Information theory | log₂ | Shannon entropy in bits |
Change of base formula: logₐ(x) = ln(x) ÷ ln(a) = log(x) ÷ log(a). Use this to calculate any base on a calculator that only has ln or log₁₀ buttons. log₂(32) = log(32) ÷ log(2) = 1.505 ÷ 0.301 = 5 ✓.