Compute natural logarithm ($\ln$), common logarithm ($\log_10$), binary logarithm ($\log_2$), or arbitrary base logarithms with bidirectional three-way solving and step-by-step mathematical proofs.
• Please provide any two values (Argument $x$, Base $b$, or Result $y$) and leave the target field blank to calculate the third.
• Base $b$ accepts numeric values or Euler's number e ($e \approx 2.7182818$) for natural logs.
Please provide any two of the following to calculate the third value: log_b(x) = y
An in-depth mathematical reference for algebra, calculus, computational physics, and computer science.
A logarithm is the inverse mathematical operation of exponentiation. In plain terms, the logarithm answers the question: “To what power (exponent) must the base $b$ be raised in order to produce the number $x$?”
Where:
Base $e \approx 2.7182818$ (Euler's Number). Universally used in calculus, continuous exponential growth/decay, physics, thermodynamics, and financial interest compounding.
ln(x) = log_e(x)Base 10. Standard scientific base used for human decimal orders of magnitude, audio decibel (dB) scales, chemistry pH measurement, and earthquake Richter scales.
log(x) = log₁₀(x)Base 2. Essential in computer science, cryptography, information entropy (Shannon bits), algorithmic time complexity ($O(\log n)$), and memory architecture.
lb(x) = log₂(x)| Rule Name | Mathematical Formula | Concrete Example |
|---|---|---|
| Product Rule | log_b(xy) = log_b(x) + log_b(y) | log₁₀(2 × 5) = log₁₀(2) + log₁₀(5) = 1 |
| Quotient Rule | log_b(x / y) = log_b(x) - log_b(y) | log₂(16 / 4) = log₂(16) - log₂(4) = 4 - 2 = 2 |
| Power Rule | log_b(xᵏ) = k · log_b(x) | log₁₀(10³) = 3 · log₁₀(10) = 3 · 1 = 3 |
| Change of Base | log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b) | log₃(81) = ln(81) / ln(3) = 4.3944 / 1.0986 = 4 |
| Reciprocal Rule | log_b(1 / x) = -log_b(x) | log₁₀(0.1) = log₁₀(1/10) = -log₁₀(10) = -1 |
| Base Switch | log_b(a) = 1 / log_a(b) | log₂(8) = 3 ⟺ log₈(2) = 1/3 |
| Identity & Zero | log_b(1) = 0, log_b(b) = 1, b^(log_b(x)) = x | ln(1) = 0, ln(e) = 1, 10^(log₁₀(50)) = 50 |
Because seismic ground motion spans many orders of magnitude, earthquake magnitudes are measured logarithmically: M = log₁₀(A / A₀). An earthquake of magnitude 7.0 releases approximately 31.6 times more energy than a 6.0!
Acidity and basicity of chemical solutions are quantified by the concentration of hydrogen ions: pH = -log₁₀[H⁺]. A solution with pH 3 has 10 times the hydrogen ion concentration of a solution with pH 4.
Human hearing perceives sound intensity logarithmically: L(dB) = 10 · log₁₀(I / I₀). A 10 dB increase represents a tenfold increase in acoustic sound wave intensity.
If the base were b = 1, then 1 raised to any power y would always equal 1 (1^y = 1). Therefore, log₁(x) would be undefined for all numbers except x = 1 (where it would have infinite solutions). To maintain well-defined mathematical functions, base 1 is strictly excluded.
The argument x must be strictly positive (x > 0) in the real number domain, because any positive base b raised to a real power always yields a positive number. However, the result y can freely be negative: for example, log₁₀(0.01) = -2 because 10^(-2) = 0.01.
In standard high school and engineering notation, log(x) denotes the common logarithm with base 10 (log₁₀ x), whereas ln(x) denotes the natural logarithm with base e (log_e x ≈ 2.7182818). In higher pure mathematics and computer science, log(x) is often used to denote natural or binary log.
Use the Change of Base Formula: log_b(x) = ln(x) / ln(b) or log₁₀(x) / log₁₀(b). For example, to calculate log₂(50), compute ln(50) / ln(2) ≈ 3.91202 / 0.693147 ≈ 5.64386.
An antilogarithm (or inverse logarithm) is simply exponentiation. If y = log_b(x), the antilog of y with base b is x = b^y.