Definition and inverse function
log_b(x) = y means bʸ = x, for base b > 0, b ≠ 1 and x > 0. The natural logarithm ln x uses base e ≈ 2.718281828; the common logarithm log x (or log₁₀ x) uses base 10; log₂ x is the binary logarithm.
The logarithm is the inverse of the exponential function: ln(eˣ) = x for every real x, and e^(ln x) = x for every x > 0. In general b^(log_b x) = x and log_b(bˣ) = x.
Sources: NIST Digital Library of Mathematical Functions: DLMF §4.2 Logarithm, Exponential, Powers: Definitions; OpenStax: College Algebra 2e §6.3 Logarithmic Functions
Logarithm rules
All rules hold for any valid base b and positive arguments x and y.
| Rule | Formula | Example |
|---|---|---|
| Product | log_b(xy) = log_b x + log_b y | log 200 = log 2 + log 100 |
| Quotient | log_b(x ÷ y) = log_b x − log_b y | ln(e³/e) = 3 − 1 = 2 |
| Power | log_b(xᵖ) = p·log_b x | log₂ 8⁵ = 5 × 3 = 15 |
| Root | log_b(ⁿ√x) = (log_b x) ÷ n | log √1000 = 3 ÷ 2 |
| Change of base | log_b x = ln x ÷ ln b = log x ÷ log b | log₂ 10 = ln 10 ÷ ln 2 ≈ 3.321928 |
| Reciprocal base | log_b x = 1 ÷ log_x b | log₈ 2 = 1 ÷ log₂ 8 = 1/3 |
| Log of 1 | log_b 1 = 0 | ln 1 = 0 |
| Log of the base | log_b b = 1 | ln e = 1 |
| Inverse | b^(log_b x) = x | e^(ln 5) = 5 |
Sources: OpenStax: College Algebra 2e §6.5 Logarithmic Properties
Special values: ln of 0, 1, e, infinity and negative numbers
| Argument | ln x | log₁₀ x | Why |
|---|---|---|---|
| x = 0 | Undefined (→ −∞) | Undefined (→ −∞) | No power of a positive base equals 0; ln x → −∞ as x → 0⁺ |
| x = 1 | 0 | 0 | b⁰ = 1 for every base |
| x = e | 1 | 0.434294 | ln e = 1 by definition of e |
| x = 10 | 2.302585 | 1 | log₁₀ 10 = 1 |
| x → ∞ | → ∞ | → ∞ | Logarithms grow without bound, but more slowly than any positive power of x |
| x < 0 | Undefined over the reals | Undefined over the reals | bʸ > 0 for every real y. In complex numbers the principal value is ln(−x) = ln x + iπ |
Derivative, integral and graph
- d/dx ln x = 1 ÷ x (x > 0), and d/dx ln|x| = 1 ÷ x (x ≠ 0)
- d/dx log_b x = 1 ÷ (x ln b)
- Chain rule: d/dx ln(u) = u′ ÷ u
- ∫ (1/x) dx = ln|x| + C
- ∫ ln x dx = x ln x − x + C
- Graph of y = ln x: defined only for x > 0, passes through (1, 0) and (e, 1), increases everywhere, is concave down, and has the y-axis as a vertical asymptote. y = log_b x has the same shape for any base b > 1 and passes through (1, 0) and (b, 1). It is the mirror image of y = bˣ in the line y = x.
Sources: OpenStax: Calculus Volume 1 §3.9 Derivatives of Exponential and Logarithmic Functions; NIST Digital Library of Mathematical Functions: DLMF §4.10 Logarithm, Exponential, Powers: Integrals; NIST Digital Library of Mathematical Functions: DLMF §4.3 Logarithm, Exponential, Powers: Graphics
Logarithm table
Natural, common and binary logarithms of selected numbers, computed to six decimal places.
| x | ln x | log₁₀ x | log₂ x |
|---|---|---|---|
| 0.1 | −2.302585 | −1.000000 | −3.321928 |
| 0.5 | −0.693147 | −0.301030 | −1.000000 |
| 1 | 0.000000 | 0.000000 | 0.000000 |
| 2 | 0.693147 | 0.301030 | 1.000000 |
| 3 | 1.098612 | 0.477121 | 1.584963 |
| 4 | 1.386294 | 0.602060 | 2.000000 |
| 5 | 1.609438 | 0.698970 | 2.321928 |
| 6 | 1.791759 | 0.778151 | 2.584963 |
| 7 | 1.945910 | 0.845098 | 2.807355 |
| 8 | 2.079442 | 0.903090 | 3.000000 |
| 9 | 2.197225 | 0.954243 | 3.169925 |
| 10 | 2.302585 | 1.000000 | 3.321928 |
| 20 | 2.995732 | 1.301030 | 4.321928 |
| 50 | 3.912023 | 1.698970 | 5.643856 |
| 100 | 4.605170 | 2.000000 | 6.643856 |
| 1000 | 6.907755 | 3.000000 | 9.965784 |