# Some Properties of the Logarithmic and Exponential Functions and Their Applications in Pharmaceutical Studies Abstract: In this paper, a few properties of logarithmic and exponential functions will be reviewed by presenting problems arising from real studies. The properties are presented under 3 topics.

## Introduction

The logarithmic (ln) and the exponential (exp) functions are two widely used mathematical functions in pharmaceutical studies. The ln function is used to transform either the concentration data or PK parameters for various reasons, such as the log-down nature of concentration in the final phase or the need to create data with normal distribution for the PK parameters. After performing some statistical or mathematical procedures in the ln-scale, the results are then back-transformed to the original scale using the exp functions.

In pharmaceutical studies, these two functions have properties that can result in some interesting and sometimes unexpected observations. In this paper, we present three topics. Under each topic, various properties of the aforementioned functions are reviewed. The first topic explains how, under special circumstances, the disproportionate scaling property of ln and exp can create unusual values for the geometric and arithmetic mean of parameters. The second topic discusses 3 types of means that can be derived in pharmaceutical studies. Normally, pharmaceutical studies only present the geometric and arithmetic means.

Using mathematical tools (and history!), we prove why the geometric mean is not a proper term to refer to the geometric means! We argue that a less popular term, i.e., the Geometric Least Square Mean (GLSM), is the right term for this type of mean. In the third topic, we present an observation which can be seen in many studies: the standard error of the ln-transformed PK parameters is very close to (sometimes identical with) the coefficient of variation of the original PK parameter. We will use mathematical tools such as the Taylor and McLaurin series expansions to prove why we make this observation.