Calculadora de Logaritmos
Calcula log, ln y logaritmos en cualquier base — incluyendo el antilogaritmo
Logarithm Calculator
Calculate logarithms with any base
Calculate log base b of x
log_b(x) = ln(x) / ln(b)What is a Logarithm Calculator?
A Logarithm Calculator is a math tool that computes logarithms, which answer the question: "What power do I raise a base to in order to get a number?" For example, if 10³ = 1000, then log₁₀(1000) = 3. The logarithm tells you the exponent (power) needed to turn 10 into 1000.
Logarithms are widely used in math and science because they help work with very large or very small numbers, convert multiplication into addition, and model real-world growth and decay. They appear in fields like chemistry (pH), finance (compound growth), engineering, computer science, and statistics.
This calculator computes all three major types of logarithms simultaneously:
Supported Logarithm Types
- Common logarithm (base 10) -- log₁₀(x), often written as log(x)
- Natural logarithm (base e) -- ln(x), where e ≈ 2.71828
- Custom base logarithm -- log₂(x) -- enter any valid base b to compute log_b(x)
How to Use This Logarithm Calculator
- Enter the number (x) -- the value you want to take the logarithm of
- Enter the base (b) -- defaults to 10, but you can change it to any valid base (e.g., 2, e, 5)
- Click "Calculate" -- to compute the logarithm
- Review all three results -- the calculator shows log_b(x), ln(x), and log₁₀(x) simultaneously
- Use the result -- apply it in your equation, problem, or real-world calculation
Tips:
- For real-number results, the input x must be greater than 0
- The base b must be greater than 0 and b ≠ 1
- If your result looks unexpected, double-check whether you need log (base 10) vs ln (base e)
Logarithm Formulas
Definition of a Logarithm
log_b(x) = y means bʸ = x
The logarithm returns the exponent y that makes bʸ equal to x
Common Logarithm
log₁₀(x)
Base 10, often written as log(x)
Natural Logarithm
ln(x)
Base e, where e ≈ 2.71828
Change of Base Formula
Compute any base using log base 10 or ln:
log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b)
Convert between any bases using this identity
Useful Logarithm Rules
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(xᵏ) = k × log_b(x)
Log of 1 / Log of Base
log_b(1) = 0
log_b(b) = 1
Example Calculations
Example 1: Common Log (Base 10)
Compute: log₁₀(1000)
Reasoning: 10³ = 1000
Result: 3
Example 2: Natural Log (Base e)
Compute: ln(e²)
Reasoning: ln returns the exponent when the base is e
Result: 2
Example 3: Custom Base Log
Compute: log₂(32)
Reasoning: 2⁵ = 32
Result: 5
Example 4: Using the Change of Base Formula
Compute: log₅(125)
Direct reasoning: 5³ = 125, so log₅(125) = 3
Change of base: ln(125) / ln(5) = 4.8283 / 1.6094 = 3
Result: 3
Frequently Asked Questions
What's the difference between log and ln?
log(x) usually means base 10 (common log), while ln(x) means base e (natural log). They're both logarithms—just with different bases.
Why can't I take the logarithm of 0 or a negative number?
In real-number math, log values are only defined for x > 0. There is no real exponent that makes a positive base equal 0 or a negative number.
What base values are allowed?
The base must be greater than 0 and not equal to 1. A base of 1 would always equal 1 for any exponent, so it can't produce different outputs.
What does a logarithm output represent?
The output is the exponent. If log_b(x) = y, then bʸ = x. That's the core meaning of logarithms.
When are logarithms useful in real life?
Logs are used when quantities change multiplicatively or span wide ranges: pH in chemistry, earthquake magnitude scales, sound intensity (decibels), compound growth/interest, and many scientific models.
Want to add this logarithm calculator to your website? Get a custom embed code that matches your site's design and keeps visitors engaged.
¿Qué es un logaritmo?
Un logaritmo responde la pregunta: ¿a qué potencia hay que elevar la base para obtener este número? Por ejemplo, log₁₀(1000) = 3 porque 10³ = 1000. Los logaritmos convierten la multiplicación en suma, lo que facilita enormemente los cálculos con números muy grandes o muy pequeños. Aparecen en ciencia, ingeniería, música e informática.
Los dos logaritmos más usados son el logaritmo base 10 (escrito simplemente "log") y el logaritmo natural (base e ≈ 2.71828, escrito "ln"). Esta calculadora acepta cualquier base — incluyendo log₂, que es fundamental en informática para medir bits e información. También puedes calcular antilogaritmos, que invierten el proceso.
Cómo usar la calculadora de logaritmos
- Ingresa el número del que quieres calcular el logaritmo (debe ser un número positivo mayor que cero).
- Selecciona la base: 10 para logaritmo común, e para logaritmo natural (ln), 2 para logaritmo binario, o ingresa una base personalizada.
- Haz clic en Calcular para ver el resultado al instante.
- Para el antilogaritmo: ingresa el valor del exponente, selecciona la misma base y la calculadora devuelve el número original (b^y).
Fórmulas e identidades del logaritmo
Definición: log_b(x) = y significa b^y = x
Logaritmo común: log(x) = log_10(x)
Logaritmo natural: ln(x) = log_e(x)
Logaritmo binario: log_2(x)
Cambio de base: log_b(x) = ln(x) / ln(b)
Antilogaritmo: antilog_10(y) = 10^y
Anti-ln: e^y
Identidades clave:
log(a × b) = log(a) + log(b)
log(a / b) = log(a) − log(b)
log(a^n) = n × log(a)Valores especiales: ln(e) = 1, log(10) = 1 y log(1) = 0 para cualquier base. Los logaritmos solo están definidos para números positivos — log(0) y el logaritmo de un número negativo no existen.
Ejemplos resueltos
log₁₀(1000) = 3
Preguntamos: ¿10 elevado a qué potencia es igual a 1000? Como 10³ = 1000, la respuesta es 3. Por eso el logaritmo común de las potencias de 10 siempre da un número entero exacto.
ln(e²) = 2
El logaritmo natural deshace la función exponencial. Como ln y e son funciones inversas, ln(e²) = 2 exactamente. Esta identidad es fundamental en cálculo diferencial e integral.
log₂(32) = 5
Preguntamos: ¿2 elevado a qué potencia es igual a 32? Como 2⁵ = 32, la respuesta es 5. El logaritmo base 2 es muy usado en informática — por ejemplo, un espacio de direcciones de 32 bits requiere log₂(2³²) = 32 bits.