log₁₀ Result
2.000000
log₁₀
2.0000
ln
4.6052
log₂
6.6439
Must be a positive number (x > 0)
2.000000
4.605170
6.643856
log10(100) = ln(100) / ln(10) = 4.605170 / 2.302585 = 2.000000
Formula: log_b(x) = ln(x) / ln(b)
log_b(x) = ln(x) / ln(b)Convert a logarithm from any base to natural log (or any other base). This is the foundation for computing arbitrary-base logarithms.
Where:
b= The base of the logarithm (b > 0, b ≠ 1)x= The number to take the logarithm of (x > 0)ln= Natural logarithm (base e ≈ 2.71828)log_b(x × y) = log_b(x) + log_b(y)The logarithm of a product equals the sum of the logarithms. This turns multiplication into addition.
Where:
x= First factor (x > 0)y= Second factor (y > 0)b= Logarithm baselog_b(x^n) = n × log_b(x)The logarithm of a power equals the exponent times the logarithm of the base value. This turns exponentiation into multiplication.
Where:
x= The base value (x > 0)n= The exponent (any real number)b= Logarithm baseInputs
Result
log₁₀(100) = 2 because 10² = 100. The common logarithm asks: what power of 10 gives 100?
Inputs
Result
ln(e³) = 3 by definition. The natural logarithm base e is the inverse of the exponential function, so ln(e^x) = x.
Inputs
Result
log₂(256) = 8 because 2⁸ = 256. In computing, this means 256 = 2⁸, so 8 bits can represent 256 different values (0–255).
A logarithm answers the question: what exponent do I need to raise the base to in order to get a given number? For example, log₁₀(1000) = 3 because 10³ = 1000. Logarithms are the inverse of exponentiation.
| Expression | Value | Because |
|---|---|---|
| log₁₀(1000) | 3 | 10³ = 1000 |
| log₂(64) | 6 | 2⁶ = 64 |
| ln(e²) | 2 | e² ≈ 7.389 |
| log₅(125) | 3 | 5³ = 125 |
log (or log₁₀) uses base 10 and is common in science and engineering. ln (natural log) uses base e ≈ 2.71828 and is fundamental in calculus and continuous growth. log₂ uses base 2 and is essential in computer science and information theory.
| Type | Base | Common Use |
|---|---|---|
| log (log₁₀) | 10 | Science, pH, decibels |
| ln | e ≈ 2.718 | Calculus, growth models |
| log₂ | 2 | Computer science, bits |
The change of base formula lets you calculate any logarithm using a different base: log_b(x) = ln(x) / ln(b) or equivalently log_b(x) = log(x) / log(b). This is essential because most calculators only have log₁₀ and ln buttons.
| Calculate | Using ln | Result |
|---|---|---|
| log₃(27) | ln(27)/ln(3) = 3.296/1.099 | 3 |
| log₅(625) | ln(625)/ln(5) = 6.438/1.609 | 4 |
| log₇(343) | ln(343)/ln(7) = 5.838/1.946 | 3 |
The three main logarithm rules are the product rule (log(ab) = log(a) + log(b)), quotient rule (log(a/b) = log(a) - log(b)), and power rule (log(a^n) = n·log(a)). These simplify complex expressions into basic arithmetic.
| Rule | Formula | Example |
|---|---|---|
| Product | log(ab) = log(a)+log(b) | log(6) = log(2)+log(3) |
| Quotient | log(a/b) = log(a)-log(b) | log(5) = log(10)-log(2) |
| Power | log(a^n) = n·log(a) | log(8) = 3·log(2) |
Logarithms appear throughout science and daily life. The Richter scale measures earthquake magnitude logarithmically (each whole number is 10x more powerful). Decibels measure sound on a log scale. The pH scale for acidity is a negative log. In finance, logarithms calculate compound interest periods.
| Application | Log Type | Example |
|---|---|---|
| Richter Scale | log₁₀ | M7 = 10× M6 |
| Decibels | log₁₀ | 10·log(P/P₀) |
| pH | −log₁₀ | pH = −log[H⁺] |
| Binary Search | log₂ | O(log₂ n) steps |
log₁₀(1000) = 3, ln(e²) = 2, and log₂(256) = 8 — each logarithm answers the same structural question: what exponent applied to the base produces the given number? The notation log_b(x) = y means bʸ = x. This inverse relationship with exponentiation is what makes logarithms indispensable for solving exponential equations.
Three bases dominate applied mathematics. The common logarithm (log₁₀) underpins the Richter scale, decibel scale, and pH scale — all systems where each unit represents a 10× change. The natural logarithm (ln, base e ≈ 2.71828) appears throughout calculus because d/dx[ln(x)] = 1/x, the simplest derivative of any logarithm. The binary logarithm (log₂) quantifies information in bits: 8 bits encode 2⁸ = 256 possible values.
The change of base formula log_b(x) = ln(x) / ln(b) lets you compute any logarithm using a single base. This is the core of how our calculator handles custom bases: log₇(343) = ln(343)/ln(7) = 5.838/1.946 = 3. The Exponent Calculator can verify results by computing the reverse: 7³ = 343.
| Expression | Value | Equivalent Exponential |
|---|---|---|
| log₁₀(10,000) | 4 | 10⁴ = 10,000 |
| ln(20.086) | 3 | e³ ≈ 20.086 |
| log₂(1024) | 10 | 2¹⁰ = 1,024 |
| log₅(3125) | 5 | 5⁵ = 3,125 |
The product rule, quotient rule, and power rule convert complex expressions into simpler arithmetic. The product rule states log(ab) = log(a) + log(b), turning multiplication into addition. Before electronic calculators, scientists used log tables to multiply large numbers by adding their logarithms — a technique that made Kepler’s planetary calculations feasible in 1627.
The power rule log(aⁿ) = n × log(a) is particularly powerful. To solve 2ˣ = 1024, take log₂ of both sides: x = log₂(1024) = 10. To find how long money takes to double at 7% annual interest, solve 1.07ⁿ = 2: n = ln(2)/ln(1.07) = 0.693/0.0677 = 10.24 years. The Scientific Notation Calculator works with logarithmic principles for the same reason — scientific notation is essentially base-10 logarithmic representation.
The quotient rule log(a/b) = log(a) − log(b) completes the triad. Combined, these rules reduce expressions like log(x³y²/z) to 3log(x) + 2log(y) − log(z). This decomposition is the foundation of log-linear regression in statistics and the reason audio engineers measure sound intensity in decibels (dB = 10 × log₁₀(P/P₀)) rather than raw power ratios.
The Richter scale assigns earthquake magnitude as log₁₀ of seismograph amplitude. A magnitude 7 earthquake releases 31.6× more energy than magnitude 6 (10¹·⁵ per step). The 2011 Tōhoku earthquake (M9.1) released approximately 1.99 × 10¹⁷ joules — about 600 million times the energy of the Hiroshima bomb.
Decibels measure sound intensity: dB = 10 × log₁₀(P/P₀), where P₀ = 10⁻¹² watts/m² is the threshold of hearing. A 90 dB factory floor is 10× more intense than 80 dB office noise. Sustained exposure above 85 dB causes hearing damage, which is why OSHA limits industrial exposure to 90 dB for 8 hours.
The pH scale in chemistry is defined as pH = −log₁₀[H⁺], where [H⁺] is hydrogen ion concentration in moles per liter. Pure water has pH 7 ([H⁺] = 10⁻⁷ M). Stomach acid at pH 1.5 is about 316,000× more acidic than water. Each 1-unit pH decrease represents a 10× increase in acidity. The Percentage Calculator can help convert between logarithmic and percentage changes when comparing concentrations.
| Scale | Formula | Each Unit = | Range |
|---|---|---|---|
| Richter | log₁₀(amplitude) | 10× amplitude, 31.6× energy | 0–10 |
| Decibels | 10·log₁₀(P/P₀) | 10× power per 10 dB | 0–194 dB |
| pH | −log₁₀[H⁺] | 10× acidity per unit | 0–14 |
| Stellar magnitude | −2.5·log₁₀(flux) | 2.512× brightness per unit | −26 to +30 |
Binary search runs in O(log₂ n) time — searching 1 billion sorted items takes at most 30 comparisons because log₂(10⁹) ≈ 30. This efficiency makes logarithmic algorithms practical even for massive datasets. Balanced binary search trees maintain O(log n) insert, delete, and lookup by ensuring tree height stays proportional to log₂(n).
Information theory defines entropy in bits using log₂. The information content of an event with probability p is −log₂(p) bits. A fair coin flip carries 1 bit (−log₂(0.5) = 1). A fair die roll carries log₂(6) ≈ 2.585 bits. Shannon entropy H = −Σ pᵢ log₂(pᵢ) measures the average information content of a source, setting the theoretical limit on lossless compression.
Sorting algorithms have a proven lower bound of O(n log n) comparisons for comparison-based sorts. Merge sort and heap sort achieve this bound exactly. The factor of log n arises because n items can be arranged in n! ways, and log₂(n!) ≈ n log₂(n) comparisons are needed to distinguish among all possible permutations.
The number of digits in a number n equals floor(log₁₀(n)) + 1. A 64-bit integer can store values up to 2⁶⁴ − 1 ≈ 1.84 × 10¹⁹, which has floor(log₁₀(1.84 × 10¹⁹)) + 1 = 20 digits.
Last Updated: Mar 26, 2026
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