Logarithm Calculator

**Logarithm definition:** log_b(y) = x means b^x = y Where: - b = base (must be positive, ≠ 1) - y = number (must be positive) - x = exponent (any real) **Three common logarithms:** - **Common (log)**: base 10. log(100) = 2. - **Natural (ln)**: base e ≈ 2.71828. ln(e) = 1. - **Binary (log₂)**: base 2. log₂(8) = 3. **Properties:** | Property | Formula | |---|---| | Product | log(a × b) = log(a) + log(b) | | Quotient | log(a/b) = log(a) − log(b) | | Power | log(a^n) = n × log(a) | | Change of base | log_a(x) = log_b(x)/log_b(a) | | Log of 1 | log(1) = 0 | | Log of base | log_b(b) = 1 | | Log of reciprocal | log(1/a) = −log(a) | **Worked examples:** log₁₀(1000) = 3 (because 10³ = 1000) log₁₀(10) = 1 log₁₀(1) = 0 log₁₀(0.01) = -2 log₂(8) = 3 (2³ = 8) log₂(1024) = 10 (2¹⁰ = 1024) log₂(1) = 0 ln(e) = 1 (e¹ = e) ln(e²) = 2 ln(1) = 0 ln(2) ≈ 0.693 **Common log values:** | Number | log₁₀ | |---|---| | 0.001 | -3 | | 0.01 | -2 | | 0.1 | -1 | | 1 | 0 | | 2 | 0.301 | | 3 | 0.477 | | 5 | 0.699 | | 7 | 0.845 | | 10 | 1 | | 50 | 1.699 | | 100 | 2 | | 1000 | 3 | | 10,000 | 4 | | 10⁶ | 6 | | 10⁹ | 9 | **Natural log values:** | Number | ln | |---|---| | 1 | 0 | | 2 | 0.693 | | e ≈ 2.718 | 1 | | 10 | 2.303 | | e² ≈ 7.389 | 2 | | 100 | 4.605 | | 1000 | 6.908 | | 10⁶ | 13.816 | **Change of base formula:** log_a(x) = ln(x) / ln(a) = log(x) / log(a) Allows any base logarithm using natural or common log: log₃(81) = ln(81) / ln(3) = 4.394 / 1.099 = 4 **Worked example: scientific notation** How many digits in 7^100? log₁₀(7^100) = 100 × log₁₀(7) = 100 × 0.845 = 84.5 So 7^100 has 85 digits (floor + 1). **Worked example: doubling time:** Continuous growth at rate k: doubling time = ln(2)/k. For 5% growth: ln(2)/0.05 = 0.693/0.05 ≈ 13.86 years. **Earthquake magnitude (Richter):** Magnitude M = log₁₀(amplitude/A₀) Each +1 in magnitude: 10× more amplitude, ~32× more energy. M7 vs M6: 10× shaking, 32× energy released. **Sound pressure (decibels):** dB = 20 × log₁₀(p/p₀) For sound, factor 20 (not 10) because power ∝ pressure²: - Sound twice as loud: +6 dB. - Sound 10× as loud: +20 dB. - Threshold of hearing: 0 dB. - Normal conversation: ~60 dB. - Pain threshold: ~130 dB. **pH scale:** pH = -log₁₀([H⁺]) For [H⁺] = 10⁻⁷ M: pH = 7 (neutral). Each pH decrease of 1 = 10× more acidic. pH 4 (vinegar) vs pH 7 (water): 1000× more H⁺ ions. **Stellar magnitude:** m = -2.5 × log₁₀(I/I₀) Brighter star = smaller magnitude. Each step of 5 magnitudes = 100× brightness ratio. Vega defined as magnitude 0; sun is -26.7; full moon -12.7; Sirius -1.46. **Information (bits):** Information content = log₂(N) bits, where N = number of equally likely outcomes. Coin flip: log₂(2) = 1 bit. Dice: log₂(6) ≈ 2.58 bits. Card from deck: log₂(52) ≈ 5.7 bits. Shannon entropy generalizes this. **Computer science:** - **Binary search**: O(log n) comparisons. - **Merge sort**: O(n log n) operations. - **Tree height** (balanced): log₂(N). - **Memory**: bits = log₂(addresses). **Logarithm in calculus:** ln(x) integrates as 1/x: ∫(1/x) dx = ln|x| + C. Derivative of ln(x) is 1/x. Derivative of e^x is e^x. These elegant relationships make natural log essential in calculus. **Logarithm history:** John Napier introduced logarithms in 1614. Henry Briggs simplified to base 10. Log tables transformed astronomy and engineering calculations for centuries. Slide rules (1622-1970s) used logarithm-marked scales for mechanical multiplication. **Decibel calculations:** For power ratio P₂/P₁: dB = 10 × log₁₀(P₂/P₁) For voltage ratio V₂/V₁: dB = 20 × log₁₀(V₂/V₁) (Factor of 20 because power = V²/R; V² gives factor of 2 in log.) **Common confusions:** - "log" alone: base 10 in elementary math, base e in much of higher math and computer science. - "ln": always natural log, base e. - "lg": sometimes binary log (base 2), context-dependent. Always specify base when ambiguous. **Programming:** | Language | Common | Natural | Custom | |---|---|---|---| | Python | math.log10(x) | math.log(x) | math.log(x, base) | | JavaScript | Math.log10(x) | Math.log(x) | Math.log(x)/Math.log(base) | | Java | Math.log10(x) | Math.log(x) | manual | | Excel | =LOG10(x) | =LN(x) | =LOG(x, base) | | R | log10(x) | log(x) | log(x, base) | | MATLAB | log10(x) | log(x) | log2(x) for binary | Note: in many programming languages (Python, R, MATLAB), log() means natural log; log10() for common log. **Software:** - **Calculators**: log button for base 10, ln for natural. - **Spreadsheets**: dedicated functions. - **Programming**: math libraries. - **Wolfram Alpha**: any base. **Logarithmic scale (plotting):** Used to visualize data spanning many orders of magnitude: - **log scale**: equal distances on axis = equal ratios. - **semi-log plot**: one axis log, other linear. Exponential growth = straight line. - **log-log plot**: both axes log. Power laws = straight line. Example: earthquake magnitudes plotted on log scale fit a power law (Gutenberg-Richter). **Common applications:** - **Chemistry**: pH, pKa, equilibrium constants. - **Acoustics**: decibels. - **Seismology**: Richter scale. - **Astronomy**: stellar magnitudes. - **Information**: bits, entropy. - **Computer science**: algorithm analysis. - **Finance**: continuous compounding. - **Population growth**: logistic models. - **Sensory perception**: many sensory scales logarithmic (Weber-Fechner law). **Pitfalls:** - **Log of 0 or negative**: undefined (real numbers). - **Base 1**: log_1(x) undefined (1 to any power is 1). - **Implicit base**: "log" means base 10 in basic math, often base e elsewhere. - **Order of operations**: log(x+1) ≠ log(x) + log(1). - **Sign**: log of < 1 is negative. - **Domain restrictions**: log only defined for positive arguments.