A novel morphological associative memory method, abbreviated as LEMAM, is constructed by using logarithmic operator and exponential operator. The theoretical analysis shows that auto LEMAM (abbreviated as ALEMAM), which has unlimited storage capacity, one step recall, and a certain ability of resisting erosive noise or dilative noise, can ensure perfect recall memory for either perfect inputs or a certain range of noise. Hetero LEMAM (abbreviated as HLEMAM) does not guarantee perfect recall, even without any input noise. However, when meeting certain conditions, HLEMAM can also achieve perfect recall. HLEMAM contrast experiments show that, in some cases, LEMAM can produce better result. On balance, LEMAM enriches the theory and practice of morphological associative memories, and can serve as a kind of new neural computational model for research and application.