Calculate logarithms with any base, including natural log and common log
A logarithm answers the question: 'To what power must we raise the base to get this number?' If log₂(8) = 3, it means 2³ = 8. Logarithms are the inverse operation of exponentiation.
There are two commonly used logarithm bases: natural logarithm (ln) with base e ≈ 2.718, used extensively in calculus and natural sciences; and common logarithm (log₁₀) with base 10, used in engineering and the decibel scale.
Logarithms transform multiplication into addition, which historically made complex calculations possible before computers. Today, they're essential in measuring earthquake intensity (Richter scale), sound levels (decibels), pH in chemistry, and exponential growth/decay.
Base 10. Used for decibels, pH scale, Richter scale. log₁₀(100) = 2.
Base e ≈ 2.718. Used in calculus, continuous growth, physics. ln(e) = 1.
Base 2. Used in computer science for bit calculations. log₂(8) = 3.
Any positive base except 1. Converted using change of base formula.
Master these rules to simplify logarithmic expressions:
| Rule | Formula | Example | Result |
|---|---|---|---|
| Product | log(xy) = log(x) + log(y) | log(2×5) | log(10) = 1 |
| Quotient | log(x/y) = log(x) - log(y) | log(100/10) | 2 - 1 = 1 |
| Power | log(xⁿ) = n·log(x) | log(10²) | 2·log(10) = 2 |
| Root | log(√x) = log(x)/2 | log(√100) | 2/2 = 1 |
| Identity | log_b(b) = 1 | log₁₀(10) | = 1 |
| Zero | log_b(1) = 0 | log₁₀(1) | = 0 |
Earthquake magnitude is log₁₀ of amplitude. Each whole number increase = 10× more ground motion. A magnitude 6 is 10× stronger than magnitude 5.
Sound intensity measured as dB = 10·log₁₀(I/I₀). Every 10 dB increase = 10× more sound intensity. 90 dB is 10× louder than 80 dB.
pH = -log₁₀[H⁺]. Each pH unit = 10× difference in acidity. pH 4 is 10× more acidic than pH 5.
Logarithms reveal doubling time in growth. Time to double = ln(2)/growth rate ≈ 0.693/r.
Find log₁₀(1000) to see the exponent that turns base 10 into 1000.
Result3
Ask: 10 raised to what power equals 1000? Since 10³ = 10 × 10 × 10 = 1000, the answer is 3. In exponential form this reads 10³ = 1000, which is exactly what the logarithm undoes.
In real numbers, no power of a positive base produces a negative result. log(-8) is undefined in real numbers. Complex logarithms extend to negative numbers using imaginary numbers.
log typically means log₁₀ (common log), while ln is logₑ (natural log, base e). In pure mathematics, 'log' often means natural log, but in engineering and calculators, it usually means log₁₀.
Convert to exponential form: if log₂(x) = 5, then x = 2⁵ = 32. For equations like log(x) + log(x-3) = 1, combine using product rule, then convert to exponential form.
e ≈ 2.71828 is Euler's number, the base of natural logarithms. It appears naturally in compound interest, growth/decay, and calculus. It's defined as lim(1 + 1/n)ⁿ as n→∞.
Any logarithm can be rewritten using a base your calculator supports: log_b(x) = ln(x)/ln(b) = log₁₀(x)/log₁₀(b). For example, log₂(64) = ln(64)/ln(2) ≈ 4.1589/0.6931 = 6. The ratio cancels the choice of intermediate base, so picking ln or log₁₀ gives the same answer.
Anywhere quantities span many orders of magnitude. Decibels measure sound intensity, the Richter scale measures earthquake amplitude, pH measures hydrogen-ion concentration in chemistry, stellar magnitudes measure brightness in astronomy, and finance uses log returns and continuously compounded interest. Each unit on these scales represents a multiplicative, not additive, change.
log_b(0) asks 'what exponent y makes b^y = 0?' For any positive base b, b^y is always positive and only approaches 0 as y → -∞ without ever reaching it. There is no finite exponent that produces 0, so the logarithm of 0 is undefined; it is treated as a limit that diverges to negative infinity.
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