Understanding Logarithms
In mathematics, a logarithm is the inverse operation to exponentiation. It answers a very specific question: "How many of one number (the base) do we multiply to get another number?"
For example, if we ask "What is the logarithm of 10,000 to base 10?", we are actually asking "10 raised to what power equals 10,000?". Because $10 \times 10 \times 10 \times 10 = 10,000$, or $10^4 = 10,000$, the answer is exactly 4. We write this mathematically as $\log_{10}(10000) = 4$. If you need to perform the reverse calculation, use our Exponent Calculator.
Common Logarithm Bases
While a logarithm can have any positive number as a base (except 1), there are a few bases that are used so frequently in science, engineering, and computer science that they have special names and notation.
Common Log (Base 10)
Often written simply as $\log(x)$ without a base specified, it is implicitly understood to be base 10. This is widely used in engineering, the Richter scale for earthquakes, and the pH scale in chemistry.
Natural Log ($\ln$)
The natural logarithm has the base $e$ (Euler's number, approximately 2.71828). It is written as $\ln(x)$. Natural logs are essential in calculus, physics, and calculating continuous compound interest.
Binary Log (Base 2)
Written as $\log_2(x)$, the binary logarithm is the foundation of computer science and information theory. It represents how many bits are required to encode a specific number of choices.
The Change of Base Formula
Most physical calculators only have two logarithm buttons: $\log$ (base 10) and $\ln$ (base e). So, how do you calculate a logarithm with a weird base, like $\log_3(50)$? You use the Change of Base Formula.
Our calculator performs this logic automatically in the background to instantly evaluate logs of any base you input, and provides the formula breakdown in the step-by-step section.
Frequently Asked Questions (FAQs)
What is a logarithm?
A logarithm answers the question: "How many of one number do we multiply to get another number?" It is the direct inverse operation to taking an exponent. The formula $\log_b(x) = y$ is mathematically identical to $b^y = x$.
What is the natural log (ln)?
The natural logarithm, commonly written as $\ln$, is a specific type of logarithm that uses the mathematical constant $e$ (Euler's number, roughly 2.71828) as its base. It is used extensively to calculate continuous growth rates in biology and finance.
Can you take the logarithm of a negative number?
In the realm of real numbers, no. You cannot raise a positive base number to any real power and get a negative result. Therefore, taking the log of a negative number (or zero) is mathematically undefined. Our calculator will alert you to this domain error.
Why can't the base be 1?
If the base is 1, the logarithm is useless. The number 1 raised to any power is always 1 ($1^2 = 1, 1^5 = 1$). You can never multiply 1 by itself to reach a number like 10, making $\log_1(10)$ completely undefined.