Калькулятор Логарифмов

Вычисляйте логарифмы в любом основании с пошаговым решением по формуле смены основания.

Что такое Калькулятор Логарифмов?

The logarithm is one of the most powerful and versatile mathematical concepts, forming the foundation of everything from earthquake measurement to signal processing, from information theory to chemical pH. At its core, log_b(n) asks: "What power must I raise b to in order to get n?" When b = 10, this is the common logarithm; when b = e ≈ 2.718, it is the natural logarithm — the language of growth, decay, and continuous change.

This calculator handles all three standard logarithm types — log base 10, natural log (ln), and any custom base — and displays the full step-by-step solution using the change-of-base formula. Understanding this formula is key: every logarithm can be expressed as a ratio of two natural logs, making computation possible with any scientific calculator.

Формула

Definition: log_b(n) = x ↔ b^x = n

Change of base: log_b(n) = ln(n) / ln(b) = log₁₀(n) / log₁₀(b)

Logarithm rules:

Product: log_b(xy) = log_b(x) + log_b(y)

Quotient: log_b(x/y) = log_b(x) − log_b(y)

Power: log_b(xⁿ) = n × log_b(x)

Special: log_b(1) = 0, log_b(b) = 1, ln(e) = 1

Как рассчитать

Enter the number n (must be positive — logarithm of zero or negatives is undefined).
Select your base: log₁₀ for common logarithm, ln for natural logarithm, or enter a custom base.
The calculator applies the change-of-base formula: log_b(n) = ln(n) / ln(b).
Verify the result by checking: b^result ≈ n (shown as antilog).
Use the reference table to compare logarithm values across different bases.

Пример

Find log₅(125): log₅(125) = ln(125)/ln(5) = 4.8283/1.6094 = 3. Verification: 5³ = 125 ✓. This makes intuitive sense because 125 = 5³.

Объяснение ключевых терминов

Logarithm: The exponent to which a base must be raised to produce a given number
Common logarithm (log₁₀): Base-10 logarithm, used in pH and decibels
Natural logarithm (ln): Base-e logarithm, fundamental in calculus and growth models
Antilogarithm: The inverse of a logarithm — b raised to the power of the log result
Change-of-base formula: log_b(n) = ln(n)/ln(b) — converts any log to computable form
Euler's number (e): Mathematical constant ≈ 2.71828, base of the natural logarithm

Частые случаи использования

Solving exponential equations in algebra and calculus
Computing pH of a solution: pH = −log₁₀[H⁺]
Calculating decibel levels: dB = 10×log₁₀(P₁/P₀)
Analyzing Richter scale earthquake magnitudes
Computing information entropy: H = −Σ p×log₂(p)
Continuous compound interest: A = Pe^(rt)

Часто задаваемые вопросы

What is a logarithm?

A logarithm answers: "To what exponent must I raise the base to get this number?" If log_b(n) = x, then b^x = n. For example, log₁₀(1000) = 3 because 10³ = 1000.

What is the natural logarithm (ln)?

The natural logarithm (ln) uses Euler's number e ≈ 2.71828 as its base. It appears in calculus, compound interest, population growth, radioactive decay, and physics. ln(x) = log_e(x).

What is the change-of-base formula?

The change-of-base formula: log_b(n) = ln(n)/ln(b) = log₁₀(n)/log₁₀(b). This means any logarithm can be computed using only ln or log₁₀.

Why are logarithms used in science?

Logarithms compress huge ranges into manageable scales. The Richter scale, decibel scale, pH scale, and stellar magnitude all use logarithms. In computing, O(log n) algorithms are among the most efficient possible.

Калькулятор Логарифмов

Что такое Калькулятор Логарифмов?

Формула

Как рассчитать

Пример

Объяснение ключевых терминов

Частые случаи использования

Часто задаваемые вопросы

Похожие инструменты

Калькулятор Логарифмов

Что такое Калькулятор Логарифмов?

Формула

Как рассчитать

Пример

Объяснение ключевых терминов

Частые случаи использования

Часто задаваемые вопросы

Похожие инструменты