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In this work, we propose a new approximate logarithm multipliers (ALM) based on a novel error compensation scheme. The proposed hardware-efficient ALM, named HEALM, first determines the truncation width for mantissa summation in ALM. Then the error compensation or reduction is performed via a lookup table, which stores reduction factors for different regions of input operands. This is in contrast to an existing approach, in which error reduction is performed independently of the width truncation of mantissa summation. As a result, the new design will lead to more accurate result with both reduced area and power. Furthermore, different from existing approaches which will either introduce resource overheads when doing error improvement or lose accuracy when saving area and power, HEALM can improve accuracy and resource consumption at the same time. Our study shows that 8-bit HEALM can achieve up to 2.92%, 9.30%, 16.08%, 17.61% improvement in mean error, peak error, area, power consumption respectively over REALM, which is the state of art work with the same number of bits truncated. We also propose a single error coefficient mode named HEALM-TA-S, which improves the ALM design with a truncation adder (TA) for mantissa summation. Furthermore, we evaluate the proposed HEALM design in a discrete cosine transformation (DCT) application. The result shows that with different values of k, HEALM-TA can improve the image quality upon the ALM baseline by 7.8 to 17.2dB in average and HEALM-SOA can improve 2.9 to15.8dB in average, respectively. Besides, HEALM-TA and HEALM-SOA outperform all the state of artworks with k=2,3,4 on the image quality. And the single coefficient mode, HEALM-TA-S, can improve the image quality upon the baseline up to 4.1dB in average with extremely low resource consumption