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Calculadora de logaritmos

The logarithm of a number to any base, including ln and log₁₀, with change-of-base steps, or the exponent x that solves bˣ = y, exact when rational.

Actualizado Ejemplos verificados: 6

Type e for the natural logarithm
Probar
Resultado
Resultado: 10
Máximo de decimales: 12; Al más cercano; empates alejándose de cero
Valor exacto
10
Natural log of the number
6.931471805599
Base-10 log of the number
3.01029995664

log base 2 of 1024 is 10: 2 must be raised to the power 10 to give 1024.

y = log base 2 of x

05105001,0001,5002,000xy(1024, 10)
Cómo se calcula S

  1. Change of base

    log⁡21,024=ln⁡1,024ln⁡2=6.931471805599450.693147180559945=10\log_{2} 1{,}024 = \frac{\ln 1{,}024}{\ln 2} = \frac{6.93147180559945}{0.693147180559945} = 10

  2. Valor exacto

    210=1,024  ⇒  log⁡21,024=102^{10} = 1{,}024 \;\Rightarrow\; \log_{2} 1{,}024 = 10

  3. Check

    210=1,0242^{10} = 1{,}024

Acerca de Calculadora de logaritmos

The logarithm log_b x is the exponent that turns the base b into x: log₂ 1024 = 10 because 2¹⁰ = 1024. Any base can be computed from natural logarithms with the change-of-base rule, log_b x = ln x ÷ ln b, and the same rule solves an exponential equation bˣ = y, giving x = ln y ÷ ln b.

Common uses are counting doublings or halvings (base 2), orders of magnitude and the decibel and pH scales (base 10), and continuous growth and decay (base e ≈ 2.71828). Solving 3ˣ = 20 gives x = ln 20 ÷ ln 3 ≈ 2.7268.

The number must be positive, and the base positive and not 1. When the answer is a fraction it is also shown exactly, so log₄ 8 appears as 3/2 as well as 1.5. A base below 1 gives negative logarithms for numbers above 1: log₀.₅ 8 = −3.

Ejemplos resueltos

log₂ 1024

Calculate
Log of x
Base b
2
Number x
1024
Resultado
10
Valor exacto
10

Fuente de comprobación: 2¹⁰ = 1024

log₁₀ 0.001

Calculate
Log of x
Base b
10
Number x
0.001
Resultado
-3
Valor exacto
−3

Fuente de comprobación: 10⁻³ = 0.001

ln 10

Calculate
Log of x
Base b
e
Number x
10
Resultado
2.302585092994

Fuente de comprobación: Python Decimal(10).ln() = 2.302585092994045684017991454684364…

log₄ 8

Calculate
Log of x
Base b
4
Number x
8
Resultado
1.5
Valor exacto
3/2

Fuente de comprobación: 4^(3/2) = (√4)³ = 8

Preguntas

What is the difference between log and ln?

ln is the natural logarithm, with base e ≈ 2.718282, while log on most calculators means base 10. Some textbooks and programming languages use log for the natural logarithm instead; Python's math.log(10) returns 2.302585. The two differ by a constant factor, ln x = ln 10 × log₁₀ x ≈ 2.302585 × log₁₀ x, so ln 10 ≈ 2.302585 while log₁₀ 10 = 1.

How do you calculate a logarithm with a different base?

Use the change-of-base rule: log_b x = ln x ÷ ln b, or equally log₁₀ x ÷ log₁₀ b. For log₂ 1024 that is 6.931472 ÷ 0.693147 = 10. For log₄ 8 it gives 1.5, which is exact because 4^(3/2) = (√4)³ = 8. Any base works except 1, and the base must be positive.

How do you solve an exponential equation like 3ˣ = 20?

Take logarithms of both sides: x × ln 3 = ln 20, so x = ln 20 ÷ ln 3 ≈ 2.995732 ÷ 1.098612 ≈ 2.726833. The same method finds doubling times: money growing 7% a year doubles when 1.07ˣ = 2, at x = ln 2 ÷ ln 1.07 ≈ 10.24 years.

Why is the logarithm of 0 or a negative number undefined?

No real power of a positive base gives 0 or a negative number: 2ˣ is positive for every real x and only approaches 0 as x heads toward −∞. So log₂ 0 has no value and ln(−1) has no real value; in complex numbers ln(−1) = iπ. A base of 1 is excluded too, since 1ˣ = 1 for every x.

What is log base 2 used for?

log₂ x counts how many times 1 must be doubled to reach x. log₂ 1024 = 10, so 1,024 = 2¹⁰ and whole numbers from 0 to 1,023 fit in 10 bits. It also bounds halving processes: a binary search of 1,024 sorted items needs at most ⌊log₂ 1024⌋ + 1 = 11 comparisons.

¿Qué precisión tiene «Calculadora de logaritmos»?

La precisión depende de tus datos y de los supuestos del método. El cálculo decimal usa 50 cifras significativas, pero las estimaciones, los métodos numéricos y los datos de origen pueden ser menos precisos; el redondeo mostrado no elimina esos límites. Ejemplos resueltos comprobados con fuentes independientes: 6. Por ejemplo, «log₂ 1024» se comprueba con 2¹⁰ = 1024.

¿De dónde procede el método?

NIST Digital Library of Mathematical Functions §4.2 — logarithms, change of base (4.2.E17); Khan Academy — Change of base formula for logarithms.

Acerca de esta calculadora

log⁡bx=ln⁡xln⁡bbx=y  ⟺  x=ln⁡yln⁡b\begin{gathered} \log_b x = \frac{\ln x}{\ln b} \\[10pt] b^{x} = y \iff x = \frac{\ln y}{\ln b} \end{gathered}

Fuentes

  1. NIST Digital Library of Mathematical Functions §4.2 — logarithms, change of base (4.2.E17)
  2. Khan Academy — Change of base formula for logarithms

Verificado con las referencias

Esta calculadora incluye 6 ejemplos resueltos con respuestas de fuentes independientes. Forman parte del conjunto de pruebas y también puedes ejecutarlos aquí.

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