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
x0
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
x0
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.