Logarithm Calculator
Use our free Logarithm Calculator to compute logarithms for any positive number in base 10, natural log (base e), or a custom base.
Result
About Logarithms: A logarithm answers the question: "To what power must the base be raised, to produce a given number?" This calculator supports base 10, natural logarithm (base e), and any positive custom base.
Practical Applications: Useful in mathematics, physics, engineering, finance, and data analysis for exponential growth, sound intensity, and pH calculations.
What Are Logarithms?
A logarithm is the inverse operation of exponentiation. It answers the question: "To what power must a given base be raised to produce a specific number?" In mathematical notation, if b^y = x, then the logarithm base b of x equals y.
logb(x) = y means by = x
For example, log10(1000) = 3 because 10 raised to the 3rd power equals 1000. Logarithms are used extensively in science, engineering, and finance to handle quantities that span many orders of magnitude.
Fundamental Logarithm Rules
Three core rules govern how logarithms behave. Mastering these rules makes it possible to simplify and solve complex logarithmic expressions.
Product Rule: logb(x * y) = logb(x) + logb(y)
Quotient Rule: logb(x / y) = logb(x) - logb(y)
Power Rule: logb(xn) = n * logb(x)
Additionally, the Change of Base Formula lets you convert between bases: logb(x) = logk(x) / logk(b). This is particularly useful when your calculator only supports base-10 or natural logarithms.
Natural Log vs. Common Log
The two most widely used logarithm bases are 10 and e (approximately 2.71828). The common logarithm (log10, often written simply as "log") is used in everyday calculations and in fields like chemistry (pH scale) and acoustics (decibels). The natural logarithm (loge, written as "ln") is central to calculus, continuous growth models, and most branches of higher mathematics.
Common Logarithm Values
| x | log10(x) | ln(x) |
|---|---|---|
| 1 | 0 | 0 |
| 2 | 0.3010 | 0.6931 |
| e (2.718) | 0.4343 | 1 |
| 10 | 1 | 2.3026 |
| 100 | 2 | 4.6052 |
| 1000 | 3 | 6.9078 |
Real-World Applications of Logarithms
Logarithms are far from purely theoretical -- they appear in many practical contexts:
- Richter Scale (Earthquakes): The Richter scale is logarithmic. An earthquake measuring 6.0 is ten times more powerful than one measuring 5.0, meaning each whole-number increase represents a tenfold increase in measured amplitude.
- Decibels (Sound Intensity): Sound levels are measured in decibels using a logarithmic scale. A sound at 80 dB is ten times more intense than one at 70 dB. This allows a vast range of intensities to be expressed in manageable numbers.
- pH Scale (Chemistry): The pH of a solution is defined as pH = -log10[H+], where [H+] is the hydrogen ion concentration. A pH change of 1 unit represents a tenfold change in acidity.
- Compound Interest (Finance): The natural logarithm is used to calculate the time required for an investment to reach a target value under continuous compounding.