7 inverse functions. 7.3 logarithmic functions inverse functions in this section, we will learn...
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7INVERSE FUNCTIONSINVERSE FUNCTIONS
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7.3Logarithmic Functions
INVERSE FUNCTIONS
In this section, we will learn about:
Logarithmic functions and natural logarithms.
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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
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If we use the formulation of an inverse
function given by (7.1.3),
then we have:
1 ( ) ( )f x y f y x
log ya x y a x
Definition 1LOGARITHMIC FUNCTIONS
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Thus, if x > 0, then logax is the exponent
to which the base a must be raised
to give x.
LOGARITHMIC FUNCTIONS
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Evaluate:
(a) log381
(b) log255
(c) log100.001
LOGARITHMIC FUNCTIONS Example 1
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(a) log381 = 4 since 34 = 81
(b) log255 = ½ since 251/2 = 5
(c) log100.001 = -3 since 10-3 = 0.001
LOGARITHMIC FUNCTIONS Example 1
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The cancellation equations (Equations 4
in Section 7.1), when applied to the functions
f(x) = ax and f -1(x) = logax, become:
log
log ( ) for every
for every 0
a
xa
x
a x x
a x x
LOGARITHMIC FUNCTIONS Definition 2
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The logarithmic function loga has
domain 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.
(0, )
LOGARITHMIC FUNCTIONS
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The figure shows the case where
a > 1.
The most important logarithmic functions have base a > 1.
LOGARITHMIC FUNCTIONS
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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
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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
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The following theorem
summarizes the properties
of logarithmic functions.
LOGARITHMIC FUNCTIONS
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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
PROPERTIES OF LOGARITHMS Theorem 3
1. log ( ) log log
2. log log log
3. log ( ) log
a a a
a a a
ra a
xy x y
xx y
y
x r x
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Properties 1, 2, and 3 follow from the
corresponding properties of exponential
functions given in Section 7.2
PROPERTIES OF LOGARITHMS
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Use the properties of logarithms
in Theorem 3 to evaluate:
(a) log42 + log432
(b) log280 - log25
Example 2PROPERTIES OF LOGARITHMS
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Using Property 1 in Theorem 3,
we have:
This is because 43 = 64.
4 4 4
4
log 2 log 32 log 2 32
log 64
3
Example 2 aPROPERTIES OF LOGARITHMS
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Using Property 2, we have:
This is because 24 = 16.
2 2 2
2
80log 80 log 5 log
5
log 16
4
Example 2 bPROPERTIES OF LOGARITHMS
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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
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If a > 1, then
In particular, the y-axis is a vertical asymptote of the curve y = logax.
LIMITS OF LOGARITHMS Equation 4
0lim log and lim loga ax x
x x
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As x → 0, we know that t = tan2x → tan20 = 0and the values of t are positive.
Hence, by Equation 4 with a = 10 > 1, we have:
LIMITS OF LOGARITHMS Example 3
2100
Find lim log tan .x
x
210 100 0
lim log tan lim logx t
x t
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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
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The logarithm with base e is called
the natural logarithm and has a special
notation:
log lne x x
NATURAL LOGARITHMS
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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 yx y e x
Definitions 5 and 6NATURAL LOGARITHMS
ln(ex ) x x °
eln x x x 0
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In particular, if we set x = 1,
we get:ln 1e
NATURAL LOGARITHMS
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Find x if ln x = 5.
From (5), we see thatln x = 5 means e5 = x
Therefore, x = e5.
E. g. 4—Solution 1NATURAL LOGARITHMS
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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 1
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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 E. g. 4—Solution 2
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Solve the equation e5 - 3x = 10.
We take natural logarithms of both sides of the equation and use Definition 9:
As the natural logarithm is found on scientific calculators, we can approximate the solution—to four decimal places: x ≈ 0.8991
5 3ln( ) ln10
5 3 ln10
3 5 ln10
1(5 ln10)
3
xe
x
x
x
Example 5NATURAL LOGARITHMS
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Express as a single
logarithm.
Using Properties 3 and 1 of logarithms, we have:
12ln lna b
1/ 212ln ln ln ln
ln ln
ln( )
a b a b
a b
a b
Example 6NATURAL LOGARITHMS
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The following formula shows that
logarithms with any base can be
expressed in terms of the natural
logarithm.
NATURAL LOGARITHMS
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For any positive number a (a ≠ 1),
we have:ln
loglna
xx
a
Formula 7CHANGE OF BASE FORMULA
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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,ln
ln
xy
a
ProofCHANGE OF BASE FORMULA
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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
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Evaluate log8 5 correct to six
decimal places.
Formula 7 gives: 8
ln 5log 5 0.773976
ln8
Example 7NATURAL LOGARITHMS
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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
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In common with all other logarithmic functions
with base greater than 1, the natural
logarithm is a continuous, increasing function
defined on and the y-axis is
a vertical asymptote.
(0, )
NATURAL LOGARITHMS
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If we put a = e in Equation 4,
then we have these limits:
NATURAL LOGARITHMS Equation 8
0lim ln lim lnx x
x x
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Sketch the graph of the function
y = ln(x - 2) -1.
We start with the graph of y = ln x.
NATURAL LOGARITHMS Example 8
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Using the transformations of Section 1.3, we shift it 2 units to the right—to get the graph of y = ln(x - 2).
Example 8NATURAL LOGARITHMS
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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:
NATURAL LOGARITHMS Example 8
2
lim ln 2 1x
x
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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
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To illustrate this fact, we compare
approximate values of the functions
y = ln x and y = x½ = in the table.x
NATURAL LOGARITHMS
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We graph the functions here.
Initially, the graphs grow at comparable rates. Eventually, though, the root function far surpasses
the logarithm.
NATURAL LOGARITHMS
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In fact, we will be able to show in
Section 7.8 that:
for any positive power p.
So, for large x, the values of ln x are very small compared with xp.
NATURAL LOGARITHMS
lnlim 0
px
x
x