

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)
= 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 = ƒ(x)
= log_{2 }x determine
f ^{1}(x). 
Solution:
a) Rewrite
y = ƒ(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. 


Exponential functions

 Exponential
function 
y
= e^{x}
<=>
x = ln y,
e = 2.718281828...the
base of the natural logarithm. 

The exponential
function is inverse
of the natural logarithm
function, so that e^{ln
}^{x} = x. 
 Exponential
function 
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},
a
> 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
(a^{x})
= log_{a }(a^{x}) =
x. 
The
domain of f
(x)
= a^{x}
is the set of all real
numbers. 
The
range of the f
(x)
= a^{x}
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 xaxis
is the horizontal 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: 












Intermediate
algebra contents 



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