4.3 Logarithmic Functions
The inverse function of the exponential function with base b is called the
logarithmic function with base b.
x
Exponential Function: f  x   b
Find the inverse:
1.) Replace f  x  with y.
2.) Interchange x and y.
3.) Solve for y.
y  bx
x  by
y  logb x
Logarithmic Function
For x  0 and b  0, b  1 ,
y  logb x is equivalent to b y  x .
b: base
x: argument
Ex. Write each equation in its equivalent exponential form or logarithmic form.
a.)
Logarithmic Form
6  log 2 64
b.)
2  log9 x
c.)
log5 125  y
Exponential Form
1
125
3
64  x
3
b  343
8 y  300
53 
d.)
e.)
f.)
g.)
Ex. Evaluate each expression without using a calculator.
a.) log 7 49 
1

b.) log3
27
c.) log6 6 
F16-CA-Miller Sec. 4.3
Page 1 of 4
Prof. LANGFORD
Special Logarithms
Common Logarithm (base 10)
Natural Logarithm (base e)
Basic Properties of Logarithms
log1  0
1.) logb 1  0
log10  1
2.) logb b  1
log10 x  log x
log e x  ln x
ln1  0
ln e  1
x
3.) log b b  x
log10 x  x
ln e x  x
log x
4.) b b  x
10log x  x
e ln x  x
Ex. Evaluate each expression without using a calculator.
a.) log11 11 
b.) log 1 
c.) ln e 

8
d.) log10 

log 22
f.) 7 7 
log a 2 3
e.) 10
ln 7 x

h.) e
2
1
g.) ln 7 
e
 Graphs of Logarithmic Functions
x
Ex. Graph f  x   3 and g  x   log3 x in the same rectangular coordinate
system.

x
-2
-1
0
1
2
y = 3x
y

x
y = log3 x







 

x








Domain of f :
Domain of g :
Range of f :
Range of g :
F16-CA-Miller Sec. 4.3
Page 2 of 4
Prof. LANGFORD


x
1
Ex. Graph f  x     and g  x   log 1 x in the same rectangular coordinate
2
2
y

system.

x
-2
-1
0
1
2
y = (1/2)x

x
y = log(1/2) x






 

x







Domain of f :
Domain of g :
Range of f :
Range of g :
Characteristics of Logarithmic Graphs of the Form f  x   logb x : (p.434)
1) Domain:
Range:
2) The point that all graphs pass through:
x-intercept:
y-intercept:
3) b  1: f  x   logb x is an
function
4) 0  b  1 : f  x   logb x is an
function
5) Vertical Asymptote:
Note:
Exponential function always has a horizontal asymptote.
Logarithmic function always has a vertical asymptote.
F16-CA-Miller Sec. 4.3
Page 3 of 4
Prof. LANGFORD



 The Domain of a Logarithmic Function
The domain of a logarithmic function, f  x   logb x , is the set of all
real
numbers.
In general, the domain of f  x   logb g  x  consists of all x for which g  x   0 .
Ex. Find the domain of each logarithmic function in interval notation.
b) f  x   ln  7  3x 
a) f  x   log5  x  6
 Transformations of Logarithmic Functions
Ex. Given the graph of f  x   log x .
i) Use the transformations of this graph to graph the given function.
ii) Give equations of the asymptotes.
iii) Use the graphs to determine each function’s domain and range.
(a) g  x   log( x  2)

(b) h  x   2  log x
y








x








   
y









x



V.A.:
V.A.:
Domain:
Domain:
Range:
Range:
F16-CA-Miller Sec. 4.3
Page 4 of 4
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



Prof. LANGFORD
   