A logarithm answers the question: 'what exponent do I need to raise a specific base to, in order to get a certain number?' For example, the logarithm of 100 to base 10 (log₁₀(100)) is 2, because 10² = 100.

What is the natural logarithm (ln)?

The natural logarithm, written as 'ln', is a logarithm with a special base called 'e' (Euler's number), which is approximately 2.718. It is widely used in science, finance, and mathematics.

What is the 'change of base' formula?

The change of base formula allows you to calculate a logarithm of any base using a calculator that only has standard log functions (like log base 10 or ln). The formula is: logₐ(x) = logₑ(x) / logₑ(a). This is how our calculator works internally.

Logarithm (Log) Calculator | CalculatorHari.com

How Does This Work?

A logarithm is the inverse operation of exponentiation. It helps answer the question: "To what exponent must we raise a given base to get the number we want?"

If y = log_b(x), then b^y = x

Since most calculators only have buttons for natural log (ln) and log base 10, this calculator uses the change of base formula to find the logarithm for any base you enter.
The formula is: log_b(x) = ln(x) / ln(b).
This allows us to accurately compute the logarithm for any base using the universally available natural logarithm function.

The Surprising History of Logarithms

In the early 17th century, before computers or calculators, complex calculations in astronomy, navigation, and engineering took an immense amount of time and were highly susceptible to errors. Scottish mathematician John Napier spent two decades developing a tool to simplify this work.

The result, published in 1614, was the invention of logarithms. This groundbreaking concept turned tedious multiplication and division problems into simple addition and subtraction. Astronomer Pierre-Simon Laplace famously said that logarithms, "by shortening the labors, doubled the life of the astronomer." For nearly 300 years, log tables and slide rules (which are built on logarithmic scales) were the most essential tools for scientists and engineers, enabling complex calculations that would have otherwise been practically impossible.

Explore More Related Tools

While you're here, check out some of our other popular math and science calculators:

Exponent & Root Calculator: The inverse operation of logarithms.
Scientific Calculator: For advanced calculations with a classic interface.
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Fraction Calculator: Add, subtract, multiply, and divide fractions.
Standard Deviation Calculator: A key tool for statistical analysis.
Unit Converter: Quickly convert between various units of measurement.
Compound Interest Calculator: See logarithms in action in finance.
Age Calculator: Find the exact age or time between two dates.
GPA Calculator: An essential tool for students.
Quadratic Equation Solver: Find the roots of quadratic equations.

Frequently Asked Questions (FAQ)

What are the common bases for logarithms?

The most common bases are: Base 10 (the "common logarithm," written as log) and Base *e* (the "natural logarithm," written as ln), where *e* is Euler's number (~2.718). In computer science, Base 2 is also very common. This calculator handles any valid base.

Why can't I calculate the logarithm of a negative number?

You cannot take the logarithm of a negative number or zero. This is because there is no real number exponent that you can raise a positive base to that will result in a negative number or zero. For example, 10^y can never be -100.

What is the base of the natural log (ln)?

The natural logarithm (ln) has a base of *e* (Euler's number), which is approximately 2.71828. To calculate the natural log with this calculator, simply enter 'e' in the base field.

Logarithm Calculator

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