Purpose This paper aims to seek purely analytical results relying on the physical parameters including the temperature jump parameter. Design/methodology/approach The exponential fin profiles and heat transfer enhancement influenced by a temperature jump at the base are the main targets of this paper. Findings The introduced temperature slip at the base penetrates through the surface of the fin and reorganizes the distribution of temperature all over the surface. The overall impact of the temperature jump on the fin efficiency is such that it acts to lower the fin efficiency. However, the efficiency of the exponential fin is increasing for growing shape exponential fins as compared to the rectangular and decaying ones. Hence, exponential fins amenable to certain temperature jump has significance in technological cooling processes. Finally, the optimum dimensions regarding the base fin thickness and the fin length of the exponential profiles are assessed by means of optimizing the base heat transfer rate given a cross-sectional area. Originality/value Exact solutions are provided for optimum exponential type fins subjected to a temperature jump. The optimum dimensions regarding the base fin thickness and the fin length of the exponential profiles are assessed.