Logarithm Calculator
Calculate logarithms with different bases
How to Use This Logarithm Calculator
Enter the number
Input any positive number to calculate its logarithm. Logarithms are only defined for positive real numbers.
Choose the base (optional)
Enter a custom base, or leave the default of 10 for common logarithms. The calculator also shows the natural log (base e).
View all results
Get the natural logarithm (ln), common logarithm (log10), and your custom base logarithm in one calculation.
Common Logarithm Values
| Number (x) | log10(x) | ln(x) | log2(x) |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 0.301030 | 0.693147 | 1 |
| 10 | 1 | 2.302585 | 3.321928 |
| 100 | 2 | 4.605170 | 6.643856 |
| 1,000 | 3 | 6.907755 | 9.965784 |
| 1,000,000 | 6 | 13.815511 | 19.931569 |
Note: log10(x) is the common logarithm, ln(x) is the natural logarithm (base e ≈ 2.718), and log2(x) is the binary logarithm.
Understanding Logarithms
A logarithm answers the question: "To what power must I raise the base to get this number?" For example, log base 10 of 100 equals 2, because 10 raised to the power of 2 equals 100. In mathematical notation: log10(100) = 2 because 10^2 = 100.
The natural logarithm uses Euler's number e (approximately 2.718) as its base. Natural logs appear throughout science and engineering because e describes continuous growth. The natural log of x is written as ln(x) or loge(x).
Logarithms convert multiplication into addition, which made them invaluable before calculators existed. Scientists used log tables and slide rules to simplify complex calculations. Today, logarithms remain essential in fields like acoustics (decibels), chemistry (pH), seismology (Richter scale), and information theory (bits and bytes).
Logarithm Properties
Product Rule
logb(x × y) = logb(x) + logb(y)
The log of a product equals the sum of the logs. Example: log10(100 × 1000) = log10(100) + log10(1000) = 2 + 3 = 5
Quotient Rule
logb(x / y) = logb(x) - logb(y)
The log of a quotient equals the difference of the logs. Example: log10(1000 / 100) = log10(1000) - log10(100) = 3 - 2 = 1
Power Rule
logb(x^n) = n × logb(x)
The log of a power equals the exponent times the log. Example: log10(100^3) = 3 × log10(100) = 3 × 2 = 6
Change of Base Formula
logb(x) = logc(x) / logc(b)
Convert between bases using this formula. Most calculators only have log10 and ln, so use this to find other bases. Example: log2(100) = log10(100) / log10(2) = 2 / 0.301 ≈ 6.64
Frequently Asked Questions
What is the difference between log and ln?
"log" without a specified base usually means log base 10 (common logarithm). "ln" means log base e (natural logarithm), where e ≈ 2.718. Common logs are used in engineering and science applications like decibels and pH. Natural logs appear in calculus, population growth models, and continuous compounding interest.
Can logarithms be negative?
The result of a logarithm can be negative. For example, log10(0.1) = -1 because 10^(-1) = 0.1. However, you cannot take the logarithm of a negative number or zero in the real number system. There's no real power of 10 that gives you -100 or 0.
Why are logarithms useful?
Logarithms compress huge ranges of values into manageable numbers. The Richter scale uses logs so an earthquake of magnitude 8 is 10 times stronger than magnitude 7. Decibels use logs to represent sound intensity. pH measures acidity on a log scale. Without logs, we'd need unwieldy numbers to describe these phenomena.
What is log base 2 used for?
Binary logarithms (base 2) are fundamental in computer science. They tell you how many bits are needed to represent a number. For example, log2(256) = 8, meaning you need 8 bits to represent 256 different values. Binary logs also appear in algorithm analysis, data compression, and information theory.
How do I calculate logarithms without a calculator?
Memorize key values like log10(2) ≈ 0.301 and log10(3) ≈ 0.477. Use logarithm properties to break down complex numbers. For example, log10(6) = log10(2 × 3) = log10(2) + log10(3) ≈ 0.301 + 0.477 = 0.778. Log tables were the standard tool before electronic calculators.
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