EXPONENTIAL and LOGARITHMIC FUNCTIONS 

LAWS OF EXPONENTS 

In the following p, q are real numbers, a, b are positive numbers and m, n are positive integers. 

LOGARITHMS AND ANTILOGARITHMS


If a^{p} = N where a ≠ 0 or 1, then p = log_{a}N is called the logarithm of N to the base a. The number N = a^{p} is called the antilogarithm of p to the base a, written antilog_{a} p. 

LAWS OF LOGARITHMS 



COMMON LOGARITHMS AND ANTILOGARITHMS 

Common logarithms and antilogarithms [also called Briggsian] are those in which the base a — 10. The common logarithm of N is denoted by log_{10} N or briefly log N. For tables of common logarithms and antilogarithms  


Natural logarithms and antilogarithms [also called Napierian] are those in which the base a = e = 2.71828 18.... The natural logarithm of N is denoted by logc N or In N. For tables of natural logarithms. For tables of natural antilogarithms [i.e. tables giving ex for values of x]  
CHANGE OF BASE OF LOGARITHMS 

The relationship between logarithms of a number N to different bases a and b is given by log_{e} N = In N = 2.30258 50929 94... log_{10}N 

RELATIONSHIP BETWEEN EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS


These are called Euler's identities. Here i is the imaginary unit 

PERIODICITY OF EXPONENTIAL FUNCTIONS 

_{e}i(θ+2fcTr) = _{e}iθ k = integer From this it is seen that ex has period 2πi. 

POLAR FORM OF COMPLEX NUMBERS EXPRESSED AS AN EXPONENTIAL


The polar form of a complex number x + iy can be written in terms of exponentials x + iy = r(cos θ + i sin θ) = re^{i θ} 

OPERATIONS WITH COMPLEX NUMBERS IN POLAR FORM 

LOGARITHM OF A COMPLEX NUMBER


ln (re^{i θ}) = ln r + iθ + 2 kπi k = integer 