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UNIT 4 :
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.