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ALGEBRA
Exponential and logarithmic functions and equations
 ::  Exponential and logarithmic functions (are mutually) 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.
Therefore, to obtain the inverse of a function y = f (x), exchange the variables x and y writing x = f (y) and solve for y.  That is, form the composition  f (f -1(x)) = x  and solve for f -1.
Example:  Given y = f (x) = log2 determine  f -1(x).
Recall, 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.

Solution:    Rewrite  y = f (x) = log2 x  to  x = log2 y  and solve for y, which gives  y = f -1(x) = 2x.

or,  form  f (f -1(x)) = x that is,  log2 (f -1(x)) = x and solve for f -1, which gives y = f -1(x) = 2x.
Graphs of inverse functions
The graphs of a pair of inverse functions are symmetrical with respect to the line  yx.
Exponential functions
    f (x) = y = ex   <=>   x = ln y,   e = 2.718281828...where e is the base of the natural logarithm.
The exponential function is inverse of the natural logarithm function, so that  e ln x = x.
    f (x) = 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 = ebx,   where a > 0  and  b = ln a
 
The graph of the logarithmic function  y = logaxa > 0  and  for  a = ey = logex = ln x
 
The logarithmic function is inverse of the f (x) = ax since its domain and the range are respectively the range and domain of the exponential function, so that
The domain of f (x) = loga x is the set of all positive real numbers.
The range of f (x) = loga x is the set of all real numbers.
If a > 1 then  f is an increasing function and if  0 < a < 1 then  f is a decreasing function.
The graph of the logarithmic function passes through the point (1, 0). The y-axis is the vertical asymptote to the graph, as shows the above picture.
Translated logarithmic and exponential functions
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.
Example:  Given translated logarithmic function  y - 2 = log3 (x + 1)find its inverse and draw their graphs.

Solution:  Exchange variables and solve for y, that is x - 2 = log3 (y + 1) which gives y + 1 = 3x - 2.

 
 
 
 
 
 
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