Exponential and Logarithmic Functions and Equations
      Inverse functions
      Exponential functions
         Translated logarithmic and exponential functions
         Rules and properties of logarithms
         Natural logarithm, common logarithm
Exponential and logarithmic functions
Exponential and logarithmic functions are mutually inverse functions
Inverse functions
The inverse function, usually written f -1, is the function whose domain and the range are respectively the range and domain of a given function f, that is
f -1(x) = y  if and only if   f (y) = x .
Thus, the composition of the inverse function and the given function returns x, which is called the identity function, i.e., 
f -1(ƒ(x)) = x    and    f (f -1(x)) = x.
The inverse of a function undoes the procedure (or function) of the given function.
A pair of inverse functions is in inverse relation.
Example:  If given  f (x) = log2 x  then f -1(x)  = 2x  since,
   
Therefore, to obtain the inverse of a function y = f (x), exchange the variables x and y, i.e., write x = f (y) and solve for y.  Or form the composition  f (f -1 (x)) = x  and solve for  f -1.
Example:  Given y = ƒ(x) = log2 determine  f -1(x).
Solution:  a)  Rewrite  y = ƒ(x) = log2 x  to  x = log2 y  and solve for y, which gives  y = f -1(x) = 2x.
                b)  Form  f (f -1(x)) = x that is,  log2 (f -1(x)) = x and solve for f -1, which gives f -1(x) = 2x.
The graphs of a pair of inverse functions are symmetrical with respect to the line  yx.
Exponential functions
- Exponential function y = ex   <=>   x = ln y,   e = 2.718281828...the base of the natural logarithm.
     The exponential function is inverse of the natural logarithm function, so that  eln x = x.
- Exponential function y = ax   <=>   x = loga  y,   where a > 0 and a is not 1.
    The exponential function with base a is inverse of the logarithmic function, so that
The graph of the exponential function  y = ax = ebxa > 0  and  b = ln a
The exponential function is inverse of the logarithmic function since its domain and the range are respectively the range and domain of the logarithmic function, so that
 f (f -1(x)) = x  that is,  f (f -1(x)) = f (ax) =  loga (ax) = x.
The domain of  f (x) = ax is the set of all real numbers.
The range of the  f (x) = ax is the set of all positive real numbers.
If a > 1 then ƒ is an increasing function and if  0 < a < 1 then ƒ is a decreasing function.
The graph of the exponential function passes through the point (0, 1). The x-axis is the horizontal asymptote to the graph, as shows the above picture.
Translated logarithmic and exponential functions
Example:  Given translated logarithmic function  y - 2 = log3 (x + 1), find its inverse and draw their graphs.
Solution:  Exchange the variables and solve for y, that is  x - 2 = log3 (y + 1)  which gives  y + 1 = 3x - 2.
Note that         y - y0 = loga (x - x0)  represents translated logarithmic function to base a 
       and          y - y0 = a (x - x0)      represents translated exponential function with base a
     where,  x0 and  y0  are the coordinates of translations of the graph in the direction of the coordinate axes.
Rules and properties of logarithms
A logarithm is the exponent (the power) to which a base must be raised to yield a given number, that is
y = loga if   x = a y.
Examples:  
 
Notation: Common logarithms of x, log10 x (to the base 10) are often written log x, without the base explicitly indicated while, Natural logarithms, loge x (to the base e, where e = 2,718218... ), are written ln x.
Therefore,
loga ax = x loga a = 1 loga 1 = 0
Logarithms are used to simplify multiplication, division and exponentiation so that,
   loga (m n) = loga m + loga n    
  loga mn = n loga m    
Example:  Using the rules of logarithms find the value of x.
 Solution:    
 
Intermediate algebra contents
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