We have developed a new microprocessor. In contrast to existing devices, which perform real arithmetic using the floating-point system, the European Logarithmic Microprocessor uses the logarithmic number system for this purpose. This paper describes the ELM device, and compares its architecture with that of a conventional floating-point digital signal processor. We then illustrate the operation of both devices using an example from a class of recursive-least-squares algorithms. The results suggest that logarithmic arithmetic may be of particular benefit in applications with less regular processing patterns, e.g. in scalar or short vector code or triangular matrix processing, or where there is a preponderance of multiplications or significant use of division or square-root operations. These criteria appear to point to the more advanced digital adaptive filtering algorithms, and also to graphics applications. Results indicate that the logarithmic number system also offers an improvement in accuracy of around one bit.