


_{}
UNIT
5 : EXPONENTIAL & LOGARITHMIC
FUNCTIONS
LESSON 5: THE
LOGARITHMIC FUNCTION
Logarithmic Notation:
The exponential statement 2^{3 }= 8 may be rewritten in what is called logarithmic form
as follows:
log_{2}8 = 3 which
reads: the logarithm of 8, base 2,
equals 3.
Any exponential statement may be rewritten in
this form.
Example 1:
Exponential Form 
Logarithmic Form 
3^{4} = 81 

5^{2} = 25 


log_{2}32 = 5 

log_{10}1000 = 3 




Solution:
Exponential Form 
Logarithmic Form 
3^{4} = 81 
log_{3}81 = 4 
5^{2} = 25 
log_{5}25 = 2 
2^{5} = 32 
log_{2}32 = 5 
10^{3} = 1000 
log_{10}1000 = 3 




Note: When we
read log_{2}8, we ask the
question To what exponent must base 2
be raised to give 8? The answer
is of course 3 and this idea
gives rise to the following definition.
Definition: the expression log_{a}x
is defined to mean the
exponent to which base a must be raised
to give x.
The expression reads: the logarithm of x, base a.
Hence log_{10}100 means the exponent to
which base 10 must be raised to give 100.
The answer is 2, giving the
statement log_{10}100 = 2.
Hence log_{3}81 means the exponent to which
base 3 must be raised to give 81. The
answer is 4, giving the
statement log_{3}81 = 4.
The Logarithmic Function:
Logarithmic functions are
the inverses of exponential functions.
How does this come about?? See example
below.
Table of Values
& Graphs:
x 
y 
3 

2 

1 

0 
1 
1 

2 

3 
8 
4 
16 
Now form the inverse of this function by interchanging x
and y in the ordered pairs and the equation.
x 
y 

3 

2 

1 
1 
0 

1 

2 
8 
3 
16 
4 
The equation of the inverse is found by interchanging x and
y in the equation y = 2^{x}. This yields
x =
2^{y}. If we put this
equation in logarithmic notation, we obtain the logarithmic function y = log_{2}x.
The graph is a reflection in the line y = x of the
exponential function y = 2^{x} and is shown in red above.
Table of Values
& graph:
x 
y 
3 

2 

1 
3 
0 
1 
1 

2 

3 

Now form the inverse of this function by interchanging x
and y in the ordered pairs and the equation.
x 
y 

3 

2 
3 
1 
1 
0 

1 

2 

3 
The equation of the inverse is found by interchanging x and
y in the equation y = (1/3)^{x}. This yields
x =
(1/3)^{y}. If we put this
equation in logarithmic notation, we obtain the logarithmic function y = log_{(1/3)}x.
The graph is a reflection in the line y = x of the
exponential function y = (1/3)^{x} and is shown in red above.
The Natural Logarithmic Function y = log_{e}x or y
= ln x:
x 

3 

2 

1 

0 
1 
1 

2 

3 

4 

Now form the inverse of this function by interchanging x
and y in the ordered pairs above.
x 
y 

3 

2 

1 
1 
0 

1 

2 

5 

4 
The equation of the inverse is found by interchanging x and
y in the equation y = e^{x}. This yields
x =
e^{y}. If we put this
equation in logarithmic notation, we obtain the logarithmic function y = log_{e}x or y = lnx.
The graph is a reflection in the line y = x of the
exponential function y = e^{x} and is shown in red above.