Transcript Slide 1

7
INVERSE FUNCTIONS
INVERSE FUNCTIONS
7.3
Logarithmic Functions
In this section, we will learn about:
Logarithmic functions and natural logarithms.
LOGARITHMIC FUNCTIONS
If a > 0 and a ≠ 1, the exponential function
f(x) = ax is either increasing or decreasing,
so it is one-to-one.
Thus, it has an inverse function f -1, which
is called the logarithmic function with base a
and is denoted by loga.
LOGARITHMIC FUNCTIONS
Definition 1
If we use the formulation of an inverse
function given by (7.1.3),
1
f ( x)  y  f ( y)  x
then we have:
loga x  y  a  x
y
LOGARITHMIC FUNCTIONS
Thus, if x > 0, then logax is the exponent
to which the base a must be raised
to give x.
LOGARITHMIC FUNCTIONS
Evaluate:
(a) log381
(b) log255
(c) log100.001
Example 1
LOGARITHMIC FUNCTIONS
Example 1
(a) log381 = 4
since 34 = 81
(b) log255 = ½
since 251/2 = 5
(c) log100.001 = -3
since 10-3 = 0.001
LOGARITHMIC FUNCTIONS
Definition 2
The cancellation equations (Equations 4
in Section 7.1), when applied to the functions
f(x) = ax and f -1(x) = logax, become:
log a (a )  x
x
a
log a x
x
for every
x
for every
x0
LOGARITHMIC FUNCTIONS
The logarithmic function loga has
domain (0, ) and range
.
 It is continuous since it is the inverse of
a continuous function, namely, the exponential
function.
 Its graph is the reflection of the graph of y = ax
about the line y = x.
LOGARITHMIC FUNCTIONS
The figure shows the case where
a > 1.
 The most important
logarithmic functions
have base a > 1.
LOGARITHMIC FUNCTIONS
The fact that y = ax is a very rapidly
increasing function for x > 0 is reflected in the
fact that y = logax is a very slowly increasing
function for x > 1.
LOGARITHMIC FUNCTIONS
The figure shows the graphs of y = logax
with various values of the base a > 1.
Since loga1 = 0,
the graphs of all
logarithmic functions
pass through the point
(1, 0).
LOGARITHMIC FUNCTIONS
The following theorem
summarizes the properties
of logarithmic functions.
PROPERTIES OF LOGARITHMS
Theorem 3
If a > 1, the function f(x) = logax is
a one-to-one, continuous, increasing
function with domain (0, ∞) and range
 If x, y > 0 and r is any real number, then
1. log a ( xy )  log a x  log a y
x
2. log a    log a x  log a y
 y
3. log a ( x r )  r log a x
.
PROPERTIES OF LOGARITHMS
Properties 1, 2, and 3 follow from the
corresponding properties of exponential
functions given in Section 7.2
PROPERTIES OF LOGARITHMS
Example 2
Use the properties of logarithms
in Theorem 3 to evaluate:
(a) log42 + log432
(b) log280 - log25
PROPERTIES OF LOGARITHMS
Example 2 a
Using Property 1 in Theorem 3,
we have:
log 4 2  log 4 32  log 4  2  32 
 log 4 64
3
 This is because 43 = 64.
PROPERTIES OF LOGARITHMS
Example 2 b
Using Property 2, we have:
 80 
log 2 80  log 2 5  log 2  
 5 
 log 2 16
4
 This is because 24 = 16.
LIMITS OF LOGARITHMS
The limits of exponential functions given
in Section 7.2 are reflected in the following
limits of logarithmic functions.
 Compare these with
this earlier figure.
LIMITS OF LOGARITHMS
Equation 4
If a > 1, then
lim loga x   and lim loga x  
x 
x 0
 In particular, the y-axis is a vertical asymptote
of the curve y = logax.
LIMITS OF LOGARITHMS
Example 3
Find lim log10  tan x  .
2
x 0
 As x → 0, we know that t = tan2x → tan20 = 0
and the values of t are positive.
 Hence, by Equation 4 with a = 10 > 1,
we have:
2
lim log10  tan x   lim log10 t
x 0
t 0
 
NATURAL LOGARITHMS
Of all possible bases a for logarithms,
we will see in Chapter 3 that the most
convenient choice of a base is the number e,
which was defined in Section 7.2.
NATURAL LOGARITHMS
The logarithm with base e is called
the natural logarithm and has a special
notation:
loge x  ln x
NATURAL LOGARITHMS
Definitions 5 and 6
If we put a = e and replace loge with ‘ln’
in (1) and (2), then the defining properties of
the natural logarithm function become:
ln x  y  e  x
y
ln(e )  x x °
x
e
ln x
x
x0
NATURAL LOGARITHMS
In particular, if we set x = 1,
we get:
ln e  1
NATURAL LOGARITHMS
Find x if ln x = 5.
 From (5), we see that
ln x = 5 means e5 = x
 Therefore, x = e5.
E. g. 4—Solution 1
NATURAL LOGARITHMS
E. g. 4—Solution 1
If you have trouble working with the ‘ln’
notation, just replace it by loge.
Then, the equation becomes loge x = 5.
So, by the definition of logarithm, e5 = x.
NATURAL LOGARITHMS
E. g. 4—Solution 2
Start with the equation ln x = 5.
Then, apply the exponential function to both
sides of the equation: eln x = e5
 However, the second cancellation equation
in Equation 6 states that eln x = x.
 Therefore, x = e5.
NATURAL LOGARITHMS
Example 5
Solve the equation e5 - 3x = 10.
 We take natural logarithms of both sides of the
equation and use Definition 9:
5 3 x
ln(e
)  ln10
5  3x  ln10
3x  5  ln10
1
x  (5  ln10)
3
 As the natural logarithm is found on scientific
calculators, we can approximate the solution—
to four decimal places: x ≈ 0.8991
NATURAL LOGARITHMS
Example 6
Express ln a  12 ln b as a single
logarithm.
 Using Properties 3 and 1 of logarithms, we have:
ln a  12 ln b  ln a  ln b1/ 2
 ln a  ln b
 ln(a b )
NATURAL LOGARITHMS
The following formula shows that
logarithms with any base can be
expressed in terms of the natural
logarithm.
CHANGE OF BASE FORMULA
Formula 7
For any positive number a (a ≠ 1),
we have:
ln x
log a x 
ln a
CHANGE OF BASE FORMULA
Proof
Let y = logax.
 Then, from (1), we have ay = x.
 Taking natural logarithms of both sides
of this equation, we get y ln a = ln x.
 Therefore, y 
ln x
ln a
NATURAL LOGARITHMS
Scientific calculators have a key for
natural logarithms.
 So, Formula 7 enables us to use a calculator
to compute a logarithm with any base—as shown
in the following example.
 Similarly, Formula 7 allows us to graph any
logarithmic function on a graphing calculator
or computer.
NATURAL LOGARITHMS
Example 7
Evaluate log8 5 correct to six
decimal places.
ln 5
 0.773976
 Formula 7 gives: log8 5 
ln 8
NATURAL LOGARITHMS
The graphs of the exponential function y = ex
and its inverse function, the natural logarithm
function, are shown.
 As the curve y = ex
crosses the y-axis with
a slope of 1, it follows
that the reflected curve
y = ln x crosses the
x-axis with a slope of 1.
NATURAL LOGARITHMS
In common with all other logarithmic functions
with base greater than 1, the natural
logarithm is a continuous, increasing function
defined on (0, ) and the y-axis is
a vertical asymptote.
NATURAL LOGARITHMS
Equation 8
If we put a = e in Equation 4,
then we have these limits:
lim ln x  
x 
lim ln x  
x 0
NATURAL LOGARITHMS
Example 8
Sketch the graph of the function
y = ln(x - 2) -1.
 We start with the graph of y = ln x.
NATURAL LOGARITHMS
Example 8
 Using the transformations of Section 1.3,
we shift it 2 units to the right—to get the
graph of y = ln(x - 2).
NATURAL LOGARITHMS
Example 8
 Then, we shift it 1 unit downward—to get
the graph of y = ln(x - 2) -1.
 Notice that the line x = 2 is a vertical
asymptote since:
lim ln  x  2  1
x 2
 
NATURAL LOGARITHMS
We have seen that ln x → ∞ as x → ∞.
However, this happens very slowly.
 In fact, ln x grows more slowly than
any positive power of x.
NATURAL LOGARITHMS
To illustrate this fact, we compare
approximate values of the functions
y = ln x and y = x½ = x in the table.
NATURAL LOGARITHMS
We graph the functions here.
 Initially, the graphs grow at comparable rates.
 Eventually, though, the root function far surpasses
the logarithm.
NATURAL LOGARITHMS
In fact, we will be able to show in
Section 7.8 that:
ln x
lim p  0
x  x
for any positive power p.
 So, for large x, the values of ln x are very small
compared with xp.