

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)
= log_{2 }x 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 =
log_{a}
x
if x = a_{
}^{y}.

Solution:
Rewrite y = f
(x)
= log_{2 }x
to x =
log_{2 }y
and solve for y,
which gives y =
f
^{1}(x)
= 2^{x}.

or, form f
(f
^{1}(x))
= x that
is, log_{2
}(f ^{1}(x))
= x and
solve for f ^{1}, which
gives y =
f ^{1}(x)
= 2^{x}.



Graphs of inverse functions

The
graphs of a pair of inverse functions are symmetrical
with respect to the line
y
= x. 


Exponential functions 
f
(x) = y
= e^{x}
<=>
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 =
a^{x}
<=>
x = log_{a
}
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
= a^{x} = e^{bx},
where
a
> 0
and b
= ln a 


The
graph of the logarithmic
function y
= log_{a}x,
a
> 0
and for a
= e,
y
= log_{e}x = ln x 

The
logarithmic
function is
inverse of the f
(x) =
a^{x}
since its
domain and the range are respectively the range and domain of
the
exponential function,
so that 

The
domain of f
(x)
= log_{a}
x is the set of all positive
real numbers. 
The
range of f
(x)
= log_{a}
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 yaxis
is the vertical asymptote to the graph, as shows the above
picture. 


Translated logarithmic
and exponential functions 
Note
that y 
y_{0}
= log_{a }(x

x_{0})
represents translated
logarithmic function to base a 
and
y 
y_{0} =
a_{ }
^{(x

x}^{0}^{)}
represents translated
exponential function with base a 
where, x_{0}
and y_{0} are the coordinates of translations of the graph in the
direction of the coordinate axes. 


Example:
Given
translated logarithmic function y 
2 = log_{3 }(x + 1),
find
its inverse and draw their
graphs.

Solution: Exchange variables
and solve for y,
that is x 
2 = log_{3 }(y + 1) which gives y
+ 1 = 3^{x

2}.

















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