logb(x), ln(x), log₂(x), and log₁₀(x) — live as you type.
You enter a positive number x and a base b (greater than 0 and not equal to 1). The tool computes the custom-base logarithm using the change-of-base identity: log_b(x) = ln(x) / ln(b). It also displays ln(x), log₂(x), and log₁₀(x) side by side. All calculations use the browser's built-in Math.log and Math.log10 functions.
A base of 1 would make the logarithm undefined because 1 raised to any power is always 1 — it can never reach any other number. Negative and zero bases are excluded because real-valued logarithms are only defined for positive bases other than 1.
In the real number system, no power of a positive base can produce zero or a negative result. Logarithms of non-positive numbers require complex numbers, which this calculator does not handle.
It states that log_b(x) = ln(x) / ln(b), or equivalently log_b(x) = log_c(x) / log_c(b) for any valid base c. This lets you compute any logarithm using just one built-in log function.
Custom bases appear in specific domains — for example, base-12 in some number systems, base-60 in time calculations, or arbitrary bases in certain information-theory and signal-processing formulas.
Estimate only. Results reflect your inputs and standard formulas. Double-check important decisions independently.