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EXPONENTIAL & LOGARITHMIC FUNCTIONS

 

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.
In a p, p is called the exponent, a is the base and ap is called the pth power of a. The function y = az is called an exponential function.

 
 
LOGARITHMS AND ANTILOGARITHMS
 
 

If ap = N where a ≠ 0 or 1, then p = logaN is called the logarithm of N to the base a. The number N = ap is called the antilogarithm of p to the base a, written antiloga p.
Example:    Since 32 = 9 we have log3 9 = 2, antilog3 2 = 9.
The function y = loga x is called a logarithmic function.

 
 

LAWS OF LOGARITHMS

 
 
  • logaMN = loga M + loga N
  • logaM/N =loga M - loga N
  • logaMp = p loga M
 
 

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 log10 N or briefly log N. For tables of common logarithms and antilogarithms  
 

NATURAL 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

In particular

loge N = In N = 2.30258 50929 94... log10N
log10N = log N = 0.43429 44819 03... logeN

 
 
RELATIONSHIP BETWEEN EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
 
 
These are called Euler's identities. Here i is the imaginary unit
 
 

PERIODICITY OF EXPONENTIAL FUNCTIONS

 
  ei(θ+2fcTr) = eiθ 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) = rei θ
 
 

OPERATIONS WITH COMPLEX NUMBERS IN POLAR FORM

 
   
 
LOGARITHM OF A COMPLEX NUMBER
 
  ln (rei θ) = ln r + iθ + 2 kπi k = integer