The memristor, because of its controllability over a wide dynamic range of resistance, has emerged as a promising device for data storage and analog computation. A major challenge is the accurate measurement of memristance over a wide dynamic range. In this paper, a novel read-out circuit with feedback adjustment is proposed to measure and digitise input current in the range between 20nA and 2mA. The magnitude of the input currents is estimated by a 5-stage logarithmic current-to-voltage amplifier which scales a linear analog-to-digital converter. This way the least significant bit tracks the absolute input magnitude. This circuit is applicable to reading single memristor conductance, and is also preferable in analog computing where read-out accuracy is particularly critical. The circuits have been realized in Bipolar-CMOS-DMOS (BCD) Gen2 technology.