Calculate logarithms with any base, natural log (ln), and common log (log10). Includes logarithmic properties, change of base formula, and step-by-step explanations.
This logarithm calculator computes logarithms with any base, including natural logarithm (ln) and common logarithm (log10). It shows the change of base formula and logarithmic properties. All calculations happen in your browser — no data is sent to any server.
A logarithm is the inverse operation to exponentiation. log_b(x) = y means that b^y = x. In other words, the logarithm tells you what power you need to raise the base to in order to get the given number.
The natural logarithm (ln) uses the mathematical constant e (approximately 2.71828) as its base. It is widely used in mathematics, physics, and engineering because of its special properties in calculus.
The change of base formula states that log_b(x) = log_c(x) / log_c(b) for any positive base c. This allows you to calculate logarithms with any base using a calculator that only has ln or log10 functions.
The three main logarithmic properties are: (1) Product Rule: log(a×b) = log(a) + log(b); (2) Quotient Rule: log(a÷b) = log(a) - log(b); (3) Power Rule: log(aⁿ) = n × log(a). These properties are essential for simplifying logarithmic expressions.