A central goal of synthetic biology is to achieve multi-signal integration and processing in living cells for diagnostic, therapeutic and biotechnology applications. Digital logic has been used to build small-scale circuits, but other frameworks may be needed for efficient computation in the resource-limited environments of cells. Here we demonstrate that synthetic analog gene circuits can be engineered to execute sophisticated computational functions in living cells using just three transcription factors. Such synthetic analog gene circuits exploit feedback to implement logarithmically linear sensing, addition, ratiometric and power-law computations. [...] Our circuits can be composed to implement higher-order functions that are well described by both intricate biochemical models and simple mathematical functions. By exploiting analog building-block functions that are already naturally present in cells, this approach efficiently implements arithmetic operations and complex functions in the logarithmic domain