Calculate logarithms for any base, log base 10, natural log (ln), log base 2, and more. Instant results with step-by-step solutions.
Logarithms convert multiplicative relationships into additive ones, and that single property drives their usefulness across science, finance, and engineering. The compound interest formula A = Pe^(rt) uses the natural log to solve for time: t = ln(A/P) / r. To double your money at 7% continuous growth, t = ln(2) / 0.07 = 9.9 years. This is why the Rule of 72 works, it approximates ln(2) ≈ 0.693. The Richter scale, decibel scale, pH scale, and f-stop scale in photography are all logarithmic. Each unit represents a multiplicative change, not an additive one. A magnitude 7.0 earthquake releases about 31.6 times more energy than a 6.0, not 1.6 times more. Understanding this distinction prevents costly misinterpretations in science and engineering. In computer science, log base 2 determines algorithm efficiency. Binary search finds any item in a sorted list of one billion in just 30 comparisons (log₂(1,000,000,000) ≈ 30). Sorting algorithms run in O(n log n) time. Every time you interact with a search engine or streaming service, logarithmic mathematics is doing the heavy lifting.
A logarithm answers a fundamental question: to what power must we raise a base number to get a given value? Written as logb(x) = y, it means by = x. So log10(1000) = 3 because 10³ = 1,000. Logarithms were invented in 1614 by John Napier specifically to simplify complex multiplication and division, before calculators existed, mathematicians used log tables to turn multiplication into addition.
Today, logarithms appear in virtually every quantitative field. The Richter scale for earthquakes is logarithmic, a magnitude 7.0 earthquake releases about 31.6 times more energy than a magnitude 6.0 quake, not 1.6 times more. The decibel scale for sound, pH scale for acidity, and the f-stop scale in photography are all logarithmic. In finance, logarithmic returns are used instead of simple returns because they're time-additive, making portfolio math far cleaner.
Computer scientists rely on log base 2 constantly. Binary search runs in O(log₂ n) time, meaning it takes only about 33 comparisons to find a specific value among 10 billion sorted items. The SHA-256 hash function underlying Bitcoin uses properties of logarithms. Data compression algorithms like Huffman coding and information theory are built on log base 2 measurements (bits).
| Base | Notation | Name | Primary Use | Example |
|---|---|---|---|---|
| 10 | log(x) or log₁₀(x) | Common log | Engineering, pH, decibels | log(100) = 2 |
| e ≈ 2.71828 | ln(x) | Natural log | Calculus, finance, physics | ln(e) = 1 |
| 2 | log₂(x) or lb(x) | Binary log | Computer science, information | log₂(8) = 3 |
Any logarithm can be converted to any other base using the change of base formula: logb(x) = ln(x) / ln(b) = log(x) / log(b). This is how our calculator computes logarithms for any base, it uses JavaScript's built-in natural log function and divides. For example, log7(2401) = ln(2401) / ln(7) = 7.784 / 1.946 = 4.000, since 7⁴ = 2,401.
| Property | Formula | Example |
|---|---|---|
| Product rule | log(a×b) = log(a) + log(b) | log(100×10) = log(100) + log(10) = 3 |
| Quotient rule | log(a/b) = log(a) − log(b) | log(1000/10) = 3 − 1 = 2 |
| Power rule | log(aⁿ) = n × log(a) | log(10⁵) = 5 × log(10) = 5 |
| Log of 1 | logb(1) = 0 | log(1) = 0, ln(1) = 0 |
| Log of base | logb(b) = 1 | log(10) = 1, ln(e) = 1 |
| Inverse | b^(logb(x)) = x | 10^(log 5) = 5 |
Industry-standard graphing calculator with built-in log and ln functions for any base.
View on Amazon →Computes natural log, log base 10, and any-base logarithm. Best value scientific calculator.
View on Amazon →Essential for plotting logarithmic curves and exponential growth visually.
View on Amazon →Understanding where logarithms appear in real life helps contextualize what can feel like abstract math. The Richter scale is perhaps the most famous example: each whole number increase represents roughly 31.6 times more energy released. A magnitude 9.0 earthquake (2011 Tohoku, Japan) released about 1,000 times more energy than a magnitude 7.0 quake. Scientists use log scales because otherwise the chart showing a 6.0 and a 9.0 would require the 9.0 bar to be 31,623 times taller.
In finance and investing, natural logarithms are used to calculate continuously compounded interest. The formula A = Pe^(rt) where e is Euler's number gives the exact amount after continuous compounding. A $10,000 investment at 7% continuously compounded for 30 years grows to $10,000 × e^(0.07×30) = $10,000 × e^2.1 ≈ $81,451. Log returns are also preferred in portfolio analysis because log(1.10) + log(1.20) = log(1.32), meaning you can simply add daily log returns to get the period total.
In music, the frequency ratio between two notes a semitone apart is 2^(1/12) ≈ 1.0595. An octave doubles frequency (log₂ of frequency doubles). This is why a piano's layout is logarithmic, equal physical distance on the keys represents equal ratios of frequency change, not equal absolute frequency changes.
| Expression | Value | Why It Matters |
|---|---|---|
| log(2) | 0.30103 | Used to find how many digits a power of 2 has |
| log(3) | 0.47712 | Key for base-3 conversions |
| log(π) | 0.49715 | Appears in normal distribution calculations |
| ln(2) | 0.69315 | Doubling time formula: t = ln(2)/r |
| ln(10) | 2.30259 | Converts log₁₀ to ln: multiply by 2.30259 |
| log₂(10) | 3.32193 | Bits needed to represent a decimal digit |
One of the most practical uses of logarithms in personal finance is calculating doubling time. If an investment grows at rate r per period, the number of periods to double is t = ln(2) / ln(1+r) ≈ 0.6931 / r (for small r). The Rule of 72 approximates this as 72/r%, so at 8% annual return, money doubles in approximately 9 years (more precisely: ln(2)/ln(1.08) = 9.006 years). At 12%, it doubles in about 6.1 years.