Describes a methodology for designing optical preamplifiers that exhibit a logarithmic response, and, therefore operate over a wide dynamic range. A novel bilinear transimpedance preamplifier is presented that can be utilized to overcome the bandwidth limitations of extant nonlinear preamplifier designs when combined with conventional logarithmic circuitry. The bandwidth improvement is achieved by exploiting the bilinear nature of the transimpedance preamplifier to implement a photocurrent gain over a very wide dynamic range. This photocurrent amplification opposes the decreased bandwidth exhibited by logarithmic amplifiers at low input current levels, which are a requirement of sensitive optical preamplification. In addition, it is demonstrated that the bilinear transimpedance approach can simplify rejection of ambient photocurrent by comparison to a single-stage logarithmic design. This method of dynamic range extension is particularly suited to optical sensing applications, where the requirements for sensitivity and monotonicity may preclude the use of continuously variable or switched gain preamplifier designs. The methodology is validated by the presentation of experimental results from a complete implementation intended for use with multiplexed-source sensing schemes, demonstrating good logarithmic conformity, high bandwidth and sensitivity, and a dynamic range comparable to variable gain designs.