Compute a logarithm in any base.

What is a logarithm?

A logarithm answers the question "what exponent do I need to raise the base to, in order to get this number?" — log_base(x) = y means base^y = x. For example, log base 10 of 100 is 2, because 10² = 100.

How do I compute a log with any base using just ln or log10?

Using the change-of-base formula: log_base(x) = ln(x) / ln(base) (equivalently, log10(x) / log10(base)). This is exactly what this calculator computes internally whenever you pick a base — which is why it can support any base you type in, not just the couple of buttons a physical calculator gives you.

What's the difference between log, ln, and log10?

"log" by itself is context-dependent — in most everyday math and engineering it means log base 10, while some computer science contexts use it to mean log base 2. "log10" is explicitly base 10, avoiding the ambiguity. "ln" is the natural logarithm, base e (Euler's number, ≈2.71828), and shows up constantly in calculus, continuous compound growth, and probability.

Why is log undefined for zero or negative numbers?

Because no real exponent applied to a positive base ever produces zero or a negative result — a positive base raised to any real power always stays positive, no matter how large or small (even negative) the exponent gets. Since a logarithm is simply asking which exponent produces a given value, there's no real answer when that value itself isn't positive. The base must also be positive and not equal to 1, for the same underlying reason.

Worked example: log base 10 of 100: ln(100) / ln(10) ≈ 4.60517 / 2.302585 = 2, confirming 10² = 100. Switching to natural log (base e) of the same value 100 gives ln(100) ≈ 4.60517, since the change-of-base formula for ln itself is just ln(x)/ln(e) = ln(x)/1 = ln(x).