The memristor, because of its controllable over a wide dynamic range 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 currentin 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 memristorconductance, 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