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Introduction and Review
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
Next are the exponential and logarithmic functions. The most common are the natural exponential and the natural logarithm. The natural exponential is given by f (x) = ex, where e ≈ 2.718281828.… The natural logarithm is the inverse to the exponential, denoted by ln x. (One needs to be careful, because some mathematics and physics books use log to mean natural exponential, whereas many of us were first trained to use this notation to mean the common logarithm, which is the “log base 10.” Here we will use ln x for the natural logarithm.)
Economic Considerations
Published in Yogesh Jaluria, Design and Optimization of Thermal Systems, 2019
where ln represents the natural logarithm. Here, ln(1 + i/m) is expanded as a Taylor series in terms of the variable i/m and m is allowed to approach infinity. Therefore, from this equation, the sum F, for continuous compounding, is given by F=Peni
Further concepts
Published in C.W. Evans, Engineering Mathematics, 2019
Any positive number, except 1, is suitable as a base. In practice two bases are used: Base 10: this produces the common logarithmsBase e: this produces the natural logarithms (also known as Nape-rian logarithms).
Bacteriostasis and cleaning effect of trace ozone replacing personal care products
Published in Environmental Technology, 2023
Yanyan Peng, Zhou Chen, Yanzeng Li, Yuantao Wang, Chengsong Ye, Junming Xu, Shenghua Zhang
C0 represent the initial concentration of ozone (mg/L). k’ represents the first order decay rate constant of ozone (min−1). Ln represents natural logarithm. Cavg represents the arithmetic square root of the product of disinfectant concentrations at times 0 and t. N0 represents the initial bacteria concentration (CFU/mL). N represents the bacteria concentration after contact time t (CFU/mL). k represents the inactivation rate constant. t represents the contact time (min). n and m represent kinetic parameters, where m is the Hom index. x represents microbial resistance to disinfectants. k1 represents sensitivity of microorganisms to disinfectants. k2 represents sensitivity of initial concentration of microorganism to disinfectant.
Combining sensor-based measurement and modeling of PM2.5 and black carbon in assessing exposure to indoor aerosols
Published in Aerosol Science and Technology, 2019
Jennie Cox, Seung-Hyun Cho, Patrick Ryan, Kelechi Isiugo, James Ross, Steven Chillrud, Zheng Zhu, Roman Jandarov, Sergey A. Grinshpun, Tiina Reponen
The mean of the CFs developed for the PM2.5RT using the MicroPEM™ PM2.5 was 0.94 (CF1 = MicroPEM™ PM2.5/PM2.5RT) with a range of 0.17 - 4.38 and a CV of 84% (Table 3). The multiple regression model, which was selected based on the lowest AIC, included PM2.5RT values, seasons, hardwood flooring, distance to highway, number of trucks, and open windows during sampling (Equation (3)). The significant variables in the linear regression included the PM2.5RT and distance to highway (Table 4). Bolded values in Equations (3)–(5) indicate significant parameters. LN is the natural logarithm. When using the multiple regression model to predict the MicroPEM™ PM2.5 values based on home characteristic variables and PM2.5RT values, the CV was calculated to be 54% (Table 3).
Data fusion for a GPS/INS tightly coupled positioning system with equality and inequality constraints using an aggregate constraint unscented Kalman filter
Published in Journal of Spatial Science, 2020
Hang Yu, Zengke Li, Jian Wang, Houzeng Han
with the accuracy of approximation controlled by a controlling factor p, which is usually specified as a sufficiently large constant (Li 1991, Fang and Wu 2016). The mathematical notation ln stands for the natural logarithm.