Why cant I take the log of a negative number?

No positive base raised to any real power gives a negative result. Since logarithms are the inverse of exponentials, log of a negative is undefined in real numbers. Complex logarithms exist but involve imaginary numbers.

What is the log of 1?

The logarithm of 1 is always 0, regardless of base. This is because any number raised to the power 0 equals 1: 10⁰ = 1, e⁰ = 1, 2⁰ = 1. Therefore log₁₀(1) = 0, ln(1) = 0, log₂(1) = 0.

How do I change the base of a logarithm?

Use the change of base formula: log_b(x) = log(x) / log(b) = ln(x) / ln(b). This lets you calculate any log using a calculator that only has log and ln. Example: log₂(8) = log(8) / log(2) = 0.903 / 0.301 = 3.

What are logarithms used for in real life?

Logarithms measure things that vary by orders of magnitude. The Richter scale (earthquakes), decibel scale (sound), and pH scale (acidity) are all logarithmic. They also solve exponential growth problems, like finding how long until an investment doubles.

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Calculadora de Logaritmos

Resolver problemas de log con base 10, e o base personalizada

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Mode

Base

calculators.log.number

Try::

log₍10₎(100)

0.00000000

log₁₀

2.000000

4.605170

log₂

6.643856

Verification (antilog)

10^2.000000 = 100.0000

Logarithm Properties

Product Rule

log(xy) = log(x) + log(y)

Quotient Rule

log(x/y) = log(x) - log(y)

Power Rule

log(xⁿ) = n·log(x)

Change of Base

log_b(x) = log(x)/log(b)

Log of 1

log(1) = 0

Log of Base

log_b(b) = 1

calculators.log.commonLogValues

Fuentes y metodología

Fórmula: log_b(x) = y means b^y = x

Logarithm definition and calculation

Fuente: Standard mathematical logarithms

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How to Use the Log Calculator

The logarithm calculator computes logarithms with any base, including common log (base 10), natural log (ln, base e), and custom bases. Visualize the log function and explore logarithm properties interactively.

What is a Logarithm?

A logarithm answers the question: "To what power must the base be raised to get this number?"

log₁₀(100) = 2 because 10² = 100

Logarithms are the inverse of exponents.

Types of Logarithms

Common log (log or log₁₀): Base 10, used in science and engineering
Natural log (ln or logₑ): Base e ≈ 2.718, used in calculus and continuous growth
Binary log (log₂): Base 2, used in computer science

Laws of Logarithms

Product Rule: log(xy) = log(x) + log(y)
Quotient Rule: log(x/y) = log(x) - log(y)
Power Rule: log(xⁿ) = n × log(x)
Change of Base: log_b(x) = log(x) / log(b)

The Antilog (Inverse)

The antilog reverses the logarithm:

If log₁₀(x) = 2, then x = 10² = 100
If ln(x) = 3, then x = e³ ≈ 20.09

Real-World Applications

Earthquake magnitude: Richter scale (logarithmic)
Sound intensity: Decibel scale
pH levels: Acidity measurement
Compound interest: Time to reach a goal

Related Calculators

For exponent calculations, use our exponent calculator. For scientific calculations, try our scientific calculator.

Preguntas frecuentes

log (common logarithm) uses base 10 and answers "what power of 10 gives this number?" ln (natural logarithm) uses base e ≈ 2.718 and answers "what power of e gives this number?" log(100) = 2 because 10² = 100. ln(e²) = 2.