Transcript Thinking Mathematically by Robert Blitzer

4.2 Logarithmic Functions
Think back to “inverse functions”. How would you find the
inverse of an exponential function such as:
f ( x)  2 x
f ( x)  x  2
f ( x)  x for x  0
f ( x)  2 x
Or more generally:
f ( x)  b x
Definition of a Logarithmic Function
• For x > 0 and b > 0, b = 1,
• y = logb x is equivalent to by = x.
(Notice that this is the INVERSE of the exponential
function f(x) = y = bx)
• The function f (x) = logb x is the
logarithmic function with base b.
Location of Base and Exponent in
Exponential and Logarithmic Forms
Exponent
Exponent
Logarithmic form: y = logb x
Exponential Form: by = x.
Base
To convert from log to exponential form, start with the
base, b, and move clockwise across the = sign:
b to the y = x.
Base
Text Example
Write each equation in its equivalent exponential form.
a. 2 = log5 x b. 3 = logb 64 c. log3 7 = y
Solution With the fact that y = logb x means by = x,
a. 2 = log5 x means:
b. 3 = logb 64 means:
.
c. log3 7 = y or y = log3 7 means:
.
Evaluate
a. log2 16
b. log3 9
c. log25 5
Solution
Logarithmic
Expression
Question Needed for
Evaluation
Logarithmic Expression Evaluated
a. log2 16
2 to what power is
16?
b. log3 9
3 to what power is 9?
c. log25 5
25 to what power is
5?
log2 16 = ____ because
2__ = 16.
log3 9 = ____ because
3__ = 9.
log25 5 = ____ because
25___ = 5.
Basic Logarithmic Properties
Involving One
• logb b = because ____is the exponent to
which b must be raised to obtain b.
(b__ = b).
• logb 1 = because ____ is the exponent to
which b must be raised to obtain 1.
(b__ = 1).
Inverse Properties of Logarithms
For b>0 and b  1,
logb bx = xThe logarithm with base b of b raised to
a power equals that power.
b logb x = x
b raised to the logarithm with
base b of a number equals that number.
That is: since logarithmic and exponential functions
are inverse functions, if they have the SAME
BASE they “cancel each other out”.
Properties of Logarithms
General Properties
1. logb 1 = 0
2. logb b = 1
3. logb bx = x
4. b logb x = x
Common
Logarithms*
1. log 1 = 0
2. log 10 = 1
3. log 10x = x
4. 10 log x = x
* If no base is written for a log, base
10 is assumed. If it says ln, that
means the “natural log” and the
base is understood to be e.
Natural
Logarithms*
1. ln 1 = 0
2. ln e = 1
3. ln ex = x
4. e ln x = x
Ex:
log 4 4 =
log 8 1 =
ln e =
ln 1 =
3 log 3 6 =
e ln 6 =
log 5 5 3 =
2 log 2 7 =
ln e 3 =
Text Example
Graph f (x) = 2x and g(x) = log2 x in the same rectangular coordinate system.
Solution We first set up a table of coordinates for f (x) = 2x. Reversing these
coordinates gives the coordinates for the inverse function, g(x) = log2 x.
x
-2
-1
0
1
2
3
x
1/4
1/2
1
2
4
8
f (x) = 2x
1/4
1/2
1
2
4
8
g(x) = log2 x
-2
-1
0
1
2
3
Reverse coordinates.
Continued…
Graph f (x) = 2x and g(x) = log2 x in the same rectangular coordinate system.
Solution
We now plot the ordered pairs in both tables, connecting them with smooth
curves. The graph of the inverse can also be drawn by reflecting the graph
of f (x) = 2x over the line y = x.
y=x
Where is the asymptote
for the exponential
function, where is it
now for the log
function?
f (x) = 2x
6
5
4
3
-2
-1
2
-1
-2
What happened to the
exponential functions yintercept?
f (x) = log2 x
2
3
4
5
6
Characteristics of the Graphs of Logarithmic
Functions of the Form f(x) = logbx
• The x-intercept is 1. There is no y-intercept.
• The y-axis is a vertical asymptote.
• If b > 1, the function is increasing. If 0 < b < 1,
the function is decreasing.
• The graph is smooth and continuous. It has no
sharp corners or edges.
Log Graphs using Transformations
shift c up c>0
g ( x)  log b x  c
shift c down c<0
g ( x)  log b ( x  c)
shift c right c>0
shift c left c<0
g ( x)  log b ( x)
reflect about y-axis
g ( x)   log b x
reflect about x-axis
Also do p 421 # 18, 64, 78, 82, 102, if time 119
18. Write in equivalent log form: b3  343
64. Graph f(x) = logx, then use transformations
to graph g(x)= 2-logx. Find the asymptote(s),
domain, range, and x- and y- intercepts.
78. Find the domain of f(x) = log (7-x)
82. Evaluate without a calculator: log 1000.
102. Write in exponential form and solve:
log5 ( x  4)  2