In this paper, we propose an exponential multiplicative fragment of linear logic to encode and solve planning problems efficiently in STRIPS domain, that we call the Linear Planning Logic (LPL). Linear logic is a resource aware logic treating resources as single use assumptions, therefore enabling encoding and reasoning of domains with dynamic state. One of the most important examples of dynamic state domains is robotic task planning, since informational or physical states of a robot include non-monotonic characteristics. Our novel theorem prover is using the backchaining method which is suitable for logic languages like Lolli and Prolog. Additionally, we extend LPL to be able to encode non-atomic conclusions in program formulae. Following the introduction of the language, our theorem prover and its implementation, we present associated algorithmic properties through small but informative examples. Subsequently, we also present a navigation domain using the hexapod robot RHex to show LPL's operation on a real robotic planning problem. Finally, we provide comparisons of LPL with two existing linear logic theorem provers, llprover and linTAP. We show that LPL outperforms these theorem provers for planning domains.