

Exponential and Logarithmic
Functions and Equations

Inverse functions 
Logarithmic functions 
Translated logarithmic
and exponential functions 
Rules and properties of logarithms

Natural logarithm, common logarithm

Changing
the base – different logarithmic identities






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}(f
(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)
= log_{2 }x
then f ^{1}(x)
= 2^{x}
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 = f
(x)
= log_{2 }x determine
f ^{1}(x). 
Solution:
a) Rewrite
y = f
(x)
= log_{2 }x
to x =
log_{2 }y
and solve for y,
which gives y =
f
^{1}(x)
= 2^{x}. 
b) Form f
(f
^{1}(x))
= x that
is, log_{2
}(f ^{1}(x))
= x and
solve for f ^{1}, which
gives f ^{1}(x)
= 2^{x}. 

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


Logarithmic functions 
 Logarithmic
function 
y
= ln x
= log_{e} x
<=>
x = e ^{y},^{ }
where x
> 0.


The
natural logarithm
function is inverse
of the exponential
function, so that
ln(e^{x}) =
x. 
 Logarithmic
function 
y =
log_{a}
x
<=>
x = a ^{y},
where a
> 0,
a
is not 1
and x
> 0. 

The
logarithmic
function with base a
is inverse of the exponential
function, so that
log_{a}(a^{x}) =
x. 


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 exponential
function 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 
Example: Given
translated logarithmic function y 
2 = log_{3 }(x + 1), find
its inverse and draw their
graphs. 
Solution:
Exchange the variables
and solve for y,
that is x 
2 = log_{3 }(y + 1)
which gives y
+ 1 = 3^{x

2}. 

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. 

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 =
log_{a}
x
if x = a_{
}^{y}. 
Examples: 



Notation:
Common logarithms of x,
log_{10} x
(to the base 10) are often written log
x, without the base
explicitly indicated while, Natural
logarithms, log_{e}
x (to the base e,
where e = 2,718218... ), are written ln
x. 
Therefore, 
log_{a}
a^{x} =
x 

log_{a}
a = 1

log_{a} 1 = 0


Logarithms
are used to simplify multiplication, division and exponentiation
so that, 

log_{a}
(m · n) =
log_{a}
m +
log_{a}
n 





log_{a}
m^{n} =
n ·
log_{a}
m^{}






Example:
Using the rules of logarithms find the value of x. 
Solution: 





Changing the base – different
logarithmic identities

Using
the identity 



In
a similar way 




therefore, 




Similarly, 


and 











Intermediate
algebra contents 



Copyright
© 2004  2020, Nabla Ltd. All rights reserved. 