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 |
Logarithms are one of the most powerful tools in mathematics, converting multiplication into addition and exponentiation into multiplication. The logarithm base b of a number x, written log_b(x), answers the question: to what power must b be raised to produce x?
The three most commonly used logarithms are the common logarithm (log₁₀), the natural logarithm (ln, base e), and the binary logarithm (log₂). Each serves different fields: log₁₀ for scientific measurement scales, ln for calculus and continuous processes, and log₂ for computer science and information theory.
Our logarithm calculator computes all three standard bases simultaneously and supports custom bases using the change of base formula. Whether you are solving equations, analyzing data, or studying for an exam, this tool provides instant results with step-by-step breakdowns.
Last Updated: Mar 9, 2026
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