Math Calculators
Logarithm & Exponent Calculator
Calculate logarithms (log₁₀, ln, any base) and powers (x^n, e^x). Supports natural log, base-10 log, and custom base logarithms.
Result: 3
log_2(8) = 3
Logarithm basics
The logarithm answers: "to what power must I raise the base to get this number?" If b^x = y, then log_b(y) = x.
Logarithm rules
| 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(4) = log(8) − log(2) |
| Power rule | log(aⁿ) = n × log(a) | log(8) = 3 × log(2) |
| Change of base | log_b(x) = ln(x) ÷ ln(b) | log_2(8) = ln(8)/ln(2) = 3 |
Common logarithms
- log₁₀ (common log): used in pH, decibels (dB), Richter scale, and early computing.
- ln (natural log, base e ≈ 2.718): used in calculus, continuous growth, finance (compound interest), and physics.
- log₂ (binary log): used in computer science - number of bits needed, algorithm complexity (O(log n) searches).
Logarithm as the inverse of exponentiation
A logarithm answers the question: "To what power must I raise the base to get this number?" If bx = y, then logb(y) = x. Think of it as the "undo" operation for exponentiation:
- 10³ = 1,000 -> log₁₀(1,000) = 3
- 2⁸ = 256 -> log₂(256) = 8
- e¹ = e -> ln(e) = 1
Natural log in finance and science
The natural logarithm (ln) appears wherever continuous processes are modeled:
- Continuous compounding: A = Pert, where P is principal, r is the annual rate, and t is time in years. The natural log reverses this: t = ln(A/P) / r.
- Radioactive decay: N(t) = N₀e−λt. The half-life T₁/₂ = ln(2) / λ ≈ 0.693 / λ.
- Population growth: when growth rate is proportional to current population, the solution is an exponential - ln linearizes it for regression analysis.
Logarithmic scales in everyday life
Many real-world measurements span many orders of magnitude, making linear scales impractical. Logarithmic scales compress these ranges:
- Earthquake magnitude (Richter / moment magnitude): each unit increase represents a 10× increase in ground motion amplitude.
- Sound intensity (decibels): 0 dB ≈ threshold of hearing; 120 dB = jet engine (10¹² times more intense). Formula: dB = 10 × log₁₀(I / I₀).
- Star brightness (apparent magnitude): a difference of 5 magnitudes = a factor of exactly 100 in brightness (brighter stars have lower magnitudes).
- pH (acidity): pH = −log₁₀([H⁺]). Each pH unit is a 10× change in hydrogen ion concentration; pH 7 is neutral, pH 1 is strongly acidic.