Download Advanced Algebra with Trig

Advanced Functions
4.3
NAME ____________________
DATE __________ PER _____
Logarithmic Functions and Their Applications
Remember: g ( x)  b x is a one-to-one function and has an inverse function.
EX 1] Find the inverse of g ( x)  b x .
Definition of a Logarithm and a Logarithmic Function
If x  0 and b is a positive constant b  1 , then y  log b x

if and only if
The notation log b x is read “the logarithm (or log) base b of x”.

The function defined by f ( x)  log b x is a logarithmic function with base b.



This function is the inverse of g ( x)  b x .
f ( x)  log b x replaces the phrase “the power of b that produces x”.
A logarithm is an exponent!
Exponential Form and Logarithmic Form:
 The exponential form of y  log b x is b y  x .

The logarithmic form of b y  x is y  log b x .
EX 2] Write each equation in its exponential form. Logarithms are EXPONENTS!
a) 3  log 2 8
b) 2  log 10 ( x  5)
c) log e x  4
d) log b b3  3
EX 3] Write each equation in its logarithmic form.
a) 32  9
b) 53  x
c) a b  c
d) blogb 5  5
by  x.
Logarithmic Properties:
1) log b b  1
2) log b 1  0
 
3)
log b b x  x
4)
b logb x  x
EX 4] Evaluate each of the following logarithms.
a) log 8 1
b) log 5 5
 
c) log 2 24
d) 3log3 7
e) log 5 625
f) log  2 
 
5
625
16
EX 5] Graph the logarithmic function f ( x)  log 2 x .
HINT: Do Loop-d-Loop first. Then, pick values for y and figure out x.
x
2y  x
y
x, y 
2
1
x
0
1
2
EX 6] Graph the logarithmic function h( x)  log  2  x .
 
3
HINT: Rewrite the function in its exponential form.
y
x
Summary of the Properties of f ( x)  log b x
For positive real numbers b, b  1 , the function f ( x)  log b x has the following properties:
1) The function f is a one-to-one function.
Domain: __________________________ Range: __________________________
2) The graph of f is a smooth continuous curve with a x-intercept of __________ , and
the graph passes through __________.
3) If b  1 , f is an increasing function and the graph of f is asymptotic to the negative y-axis.
This means As x  ,
f ( x)  _____
as x  0 from the RIGHT ,
, and
f ( x)  _____ .
4) If 0  b  1 , f is a decreasing function and the graph of f is asymptotic to the positive y-axis.
This means As x  , f ( x)  _____ , and
as x  0 from the RIGHT , f ( x)  _____ .
y
y
f ( x)  log b x
x
f ( x)  log b x
x
0  b  1
b  1
EX 7] What is the domain of the graph of each of the following?
a)
f ( x)  log 6 ( x  3)

b) g ( x)  ln x 4  x 2

 x 

c) h( x)  log 5 
8

x


EX 8] Explain how to use the graph of f ( x)  log 4 x to produce the graph of g ( x)  log 4 ( x  3) .
y
x
EX 9] Explain how to use the graph of f ( x)  log 4 x to produce the graph of h( x)  log 4 x  3 .
y
x
Definition of Common Logarithms:
 logarithms of base 10
 EX] f ( x)  log 10 x  log x
Definition of Natural Logarithms:
 logarithms of base e
 EX] f ( x)  log e x  ln x