Calculadora de Logaritmos

Calcula log₁₀, ln, log₂ e logaritmos personalizados.

Calculadoras

Como usar Calculadora de Logaritmos

  1. 1Enter the number (must be > 0)
  2. 2See log₁₀, ln, and log₂ results instantly
  3. 3Enter a custom base to see logₙ

Sobre Calculadora de Logaritmos

The Logarithm Calculator computes all common logarithms simultaneously: the common logarithm (log base 10), the natural logarithm (ln, base e), the binary logarithm (log base 2), and a custom base logarithm for any user-defined base. Enter a single number and see all four results at once.

Logarithms are the inverse of exponentiation: if 10^x = y, then log10(y) = x. The natural logarithm uses base e (approximately 2.71828) and is fundamental in calculus, probability, and finance. Log base 2 is essential in computer science for bit complexity analysis.

All calculations run in your browser with no server calls. The tool is equally useful for chemistry pH calculations, decibel conversions, information theory problems, and general algebraic equation solving.

Principais recursos de Calculadora de Logaritmos

  • Compute log base 10 (common logarithm) instantly
  • Compute natural logarithm ln (base e = 2.71828...)
  • Compute log base 2 (binary logarithm) for computer science
  • Compute logarithm with any custom base n
  • Shows all four results simultaneously for quick comparison
  • Enforces the requirement that input must be positive (> 0)
  • Handles very large and very small positive inputs
  • Instant results without page reload

Exemplos

Logarithm of 1000 in base 10

Verify that log10(1000) = 3 because 10^3 = 1000.

Entrada

x = 1000

Saída

log10: 3 | ln: 6.908 | log2: 9.966

Custom base logarithm

Calculate log base 3 of 81.

Entrada

x = 81, custom base = 3

Saída

log3(81) = 4 (because 3^4 = 81)

Casos de uso comuns

  • Calculating pH (pH = -log10[H+]) in chemistry
  • Converting decibel levels using the log10 formula
  • Analyzing algorithm complexity using log base 2 (binary search = O(log n))
  • Solving exponential equations by taking logarithms of both sides
  • Computing compound interest time periods using natural logarithm
  • Understanding information entropy in information theory

Solução de problemas

Entering zero or a negative number as the input

Solução

Logarithms are only defined for positive real numbers. log(0) is negative infinity and log(-x) requires complex numbers. Enter only positive values greater than zero.

Entering 1 as the custom base

Solução

Logarithm base 1 is undefined because 1^x = 1 for all x, making it impossible to solve 1^x = y for any y != 1. The base must be positive and not equal to 1.

Confusing log (base 10) with ln (base e)

Solução

In mathematics, "log" without a base specified often means log base 10 in applied contexts or ln in pure math. This calculator labels them explicitly to avoid ambiguity.

Perguntas frequentes

What is ln (natural logarithm)?

ln(x) is the logarithm with base e (Euler's number, approximately 2.71828). It is the inverse of e^x. ln is used extensively in calculus, differential equations, probability, and finance.

What is the common logarithm (log10)?

log10(x) asks: to what power must 10 be raised to get x? For example, log10(1000) = 3 because 10^3 = 1000. It is used in pH calculations, decibel scales, and order-of-magnitude estimation.

What is log base 2 used for?

Log base 2 (binary logarithm) is fundamental in computer science. It tells you the number of bits needed to represent n values, and appears in the time complexity of algorithms like binary search: O(log2 n).

How do I compute log of a custom base?

Enter your number in the main input and your desired base in the custom base field. The calculator uses the change-of-base formula: log_b(x) = ln(x) / ln(b).

Why must the input be positive?

Logarithms of zero or negative numbers are not defined in the real number system. ln(0) = -infinity and ln(-x) requires complex numbers, which are beyond the scope of this calculator.

What is the relationship between log and exponents?

Logarithms and exponentiation are inverse operations. If b^y = x, then log_b(x) = y. For example, 10^3 = 1000, so log10(1000) = 3.

Can I compute log of a number between 0 and 1?

Yes. Logarithms of numbers between 0 and 1 are negative. For example, log10(0.001) = -3 because 10^(-3) = 0.001.

Is my data private?

Yes. All calculations run entirely in your browser. No values are transmitted to any server.