Understanding Logarithms

A logarithm is the mathematical operation that determines how many times a certain number, called the base, must be multiplied by itself to reach another number. In simple terms, a logarithm is the inverse operation of exponentiation. If you want to learn more about exponentiation, check out our Exponent Calculator.

The relationship is expressed as:
log_b(x) = y is equivalent to b^y = x

For example, log₂(8) = 3 because 2 must be raised to the power of 3 to get 8 (2³ = 8).

Common Types of Logarithms

While a logarithm can have any positive number as its base, a few are so common they have special names:

• Common Logarithm (log₁₀ or log): This logarithm uses base 10. It is widely used in science and engineering, especially for measurements on the Richter scale (earthquakes), pH scale (acidity), and decibel scale (sound intensity).
• Natural Logarithm (ln): This logarithm uses the special irrational number e (approximately 2.71828) as its base. The natural log is fundamental in calculus, physics, and finance to model continuous growth and decay. Problems involving this can be seen in our Half-Life Calculator.
• Binary Logarithm (log₂): This logarithm uses base 2. It is central to computer science and information theory, as it relates to the binary (base-2) system that computers use.

The Change of Base Formula

Most calculators, including our powerful Scientific Calculator, can only directly compute common and natural logs. To find a logarithm to an arbitrary base, you must use the change of base formula:

log_b(x) = log_k(x) / log_k(b)

You can use any new base 'k', but 10 or 'e' are the most convenient. Our calculator automates this process for you.

How to Use Our Calculator

1. Select the Log Type: Choose from a custom base (log_b), natural log (ln), common log (log₁₀), or binary log (log₂). The base input will update automatically.
2. Enter the Number (x): This is the number you want to find the logarithm of. It must be positive.
3. Click Calculate: The result will be displayed.

For more advanced mathematical problem-solving, browse our full collection of Math Calculators.