Logarithm Calculator
Calculate logarithms in any base: log base 10 (common log), natural log (ln), log base 2 (binary), or any custom base. Also computes antilogs and converts between bases. Step-by-step work shown for each calculation.
A — Calculate logb(x)
Change-of-base formula: log(100) = ln(100) / ln(10) = 4.60517019 / 2.30258509 = 2 Verification: 10^2 = 100
B — Antilog (Inverse Logarithm)
Antilog (inverse logarithm): If log(x) = 2, then x = 10^2 x = 10^2 x = 100 Verification: log(100) = 2
C — Change of Base
Change-of-base formula: log_b(x) = ln(x) / ln(b) log(100) = ln(100) / ln(10) = 4.60517019 / 2.30258509 = 2 log₂(100) = ln(100) / ln(2) = 4.60517019 / 0.69314718 = 6.64385619 Converting between bases: log₂(100) = log(100) / log(2) = 2 / 0.30103 = 6.64385619
What Is a Logarithm?
A logarithm is the inverse operation of exponentiation. If you raise a base b to a power x and get y, then the logarithm base b of y equals x:
If bx = y, then logb(y) = x
For example: 103 = 1000, so log10(1000) = 3. This reads as "the log base 10 of 1000 is 3." The logarithm answers: "to what power must 10 be raised to equal 1000?"
The three most common logarithm bases are:
- log (base 10) — the common logarithm, used in science and engineering. Written simply as log x.
- ln (base e) — the natural logarithm, where e ≈ 2.71828. Used in calculus, growth/decay models.
- lb (base 2) — the binary logarithm, used in computer science and information theory.
Laws of Logarithms
Logarithm laws follow directly from exponent laws. Mastering these allows you to simplify complex expressions without evaluating them numerically.
| Law | Formula | Example (base 10) |
|---|---|---|
| Product rule | logb(xy) = logb(x) + logb(y) | log(100 × 10) = log(100) + log(10) = 2 + 1 = 3 |
| Quotient rule | logb(x/y) = logb(x) − logb(y) | log(1000/10) = log(1000) − log(10) = 3 − 1 = 2 |
| Power rule | logb(xn) = n × logb(x) | log(1003) = 3 × log(100) = 3 × 2 = 6 |
| Change of base | logb(x) = ln(x) / ln(b) | log2(8) = ln(8) / ln(2) = 2.079 / 0.693 = 3 |
| Log of 1 | logb(1) = 0 | log(1) = 0, ln(1) = 0 |
| Log of base | logb(b) = 1 | log(10) = 1, ln(e) = 1 |
The Natural Logarithm (ln)
The natural logarithm uses Euler's number e ≈ 2.71828 as its base. It arises naturally in calculus: the derivative of ex is ex, and the integral of 1/x is ln(x).
Natural logs appear in:
- Compound interest: A = Pert where t is time and r is continuous growth rate
- Population growth: N(t) = N0ekt
- Radioactive decay: N(t) = N0e−λt
- Entropy in thermodynamics: S = kB ln(Ω)
Common Logarithm (log base 10)
The common logarithm (base 10) is written simply as log x. It counts the number of decimal digits minus one: log(10) = 1, log(100) = 2, log(1000) = 3.
Applications include:
- Decibels: Sound level (dB) = 10 × log10(P / P0)
- Richter scale: Each unit increase = 10× stronger earthquake
- pH scale: pH = −log10[H+]
- Stellar magnitude: Brightness differences are logarithmic
Binary Logarithm (log base 2)
The binary logarithm answers: "how many times must 2 be doubled to reach x?" It is fundamental in computer science because computers work in binary (base 2).
- log2(256) = 8 (8 bits = 1 byte)
- log2(1024) = 10 (10 bits)
- Algorithm complexity: O(log n) usually means O(log2 n)
- Information content: log2(n) bits needed to identify 1 of n equally likely events
Change of Base Formula
Calculators typically provide log (base 10) and ln (base e) buttons. To compute logb(x) for any base b, use the change-of-base formula:
logb(x) = log(x) / log(b) = ln(x) / ln(b)
Example: log5(125) = log(125) / log(5) = 2.0969 / 0.6990 = 3. Verify: 53 = 125. ✓
Frequently Asked Questions
What is a logarithm?
A logarithm answers the question: "To what power must the base be raised to produce a given number?" If b^x = y, then log_b(y) = x. For example, 10^2 = 100, so log₁₀(100) = 2. Logarithms are the inverse of exponentiation, just as subtraction is the inverse of addition.
What is the difference between log and ln?
log (written without a base) typically means log base 10 (common logarithm). ln means log base e, where e ≈ 2.71828 (natural logarithm). log₂ is log base 2 (binary logarithm, used in computer science). The three are related by the change-of-base formula: log_b(x) = ln(x) / ln(b).
What are the laws of logarithms?
The main log laws are: Product rule: log_b(xy) = log_b(x) + log_b(y). Quotient rule: log_b(x/y) = log_b(x) − log_b(y). Power rule: log_b(x^n) = n × log_b(x). Change of base: log_b(x) = log_c(x) / log_c(b) for any valid base c. These mirror the laws of exponents since logs are inverse exponentials.
What is the antilogarithm (antilog)?
The antilogarithm is the inverse operation of a logarithm. If log_b(y) = x, then the antilog base b of x is y = b^x. For common logs: antilog(2) = 10² = 100. For natural logs: antilog(1) = e¹ = e ≈ 2.718. Use Section B of this calculator to compute antilogs.
Where are logarithms used in real life?
Logarithms appear in sound measurement (decibels: dB = 10 log₁₀(P/P₀)), earthquake magnitude (Richter scale), pH in chemistry (pH = −log₁₀[H⁺]), music (musical intervals are logarithmic), compound interest calculations, information theory (bits = log₂ of possibilities), and exponential growth/decay problems in biology and physics.
Related Calculators
- Exponent Calculator — the inverse of logarithms
- Scientific Calculator — full expression parser with log and ln buttons
- Standard Deviation Calculator — uses logarithms in statistical distributions