a.Let tbe the number of years since 2000, and Cthe capacity (in MW). Find functions of the form
and that fit the data at 2001 and 2009.
b.Use a graphing calculator to plot the data in the table and to graph the three functions found in part a. Which function best fits the data?
c.If you have a graphing calculator or computer program with an exponential regression feature, use it to find an exponential function that approximately fits the data in the table. How does this answer compare with the answer to part b?
d.Using the three functions from part b and the function from part c, predict the total world wind capacity in 2010. Compare these with the World Wind Energy Association’s prediction of 203,500.
C5abt C5mt1b,C5at21b,
Year Capacity (MW)
2001 24,322
2002 31,181
2003 39,295
2004 47,693
2005 58,024
2006 74,122
2007 93,930
2008 120,903 2009 159,213
YOUR TURN ANSWERS
1. 9/2 2. $772.97
3. $907.43 2
Logarithmic Functions
With an inflation rate averaging 5% per year, how long will it take for prices to double?
2.5
APPLY IT
The number of years it will take for prices to double under given conditions is called the doubling time.For $1 to double (become $2) in t years, assuming 5% annual com- pounding, means that
becomes
or
This equation would be easier to solve if the variable were not in the exponent. Loga- rithmsare defined for just this purpose. In Example 8, we will use logarithms to answer the question posed above.
25 11.052t. 251a11 0.05
1 b11t2 A5Pa11 r
mbmt
(Read as “yis the logarithm of xto the base a.”) For example, the exponential statement can be translated into the logarithmic statement Also, in the problem discussed above, can be rewritten with this definition as
A logarithm is an exponent: is the exponent used with the base ato get x.
Equivalent Expressions
This example shows the same statements written in both exponential and logarithmic forms.
Exponential Form Logarithmic Form (a)
(b) (c) (d) (e)
(f ) TRY YOUR TURN 1
Evaluating Logarithms Evaluate each of the following logarithms.
(a)
SOLUTION We seek a number xsuch that Since we conclude that (b)
SOLUTION We seek a number xsuch that Since is positive for all real num- bers x, we conclude that is undefined. (Actually, can be defined if we use complex numbers, but in this textbook, we restrict ourselves to real numbers.) (c)
SOLUTION We know that and so and
Therefore, must be somewhere between 2 and 3. We will find a more accurate
answer in Example 4. TRY YOUR TURN 2
Logarithmic Functions For a given positive value of x, the definition of loga- rithm leads to exactly one value of y, so defines the logarithmic functionof base a(the base amust be positive, with a2y1).5loga x
log5 80 52525 535125, log5 2552 log5 12553.
log5 80
log21282 log21282 2x5 28. 2x log21282
log4 6453. 4x564. 43564,
log4 64
loge 150 e051
log211/1625 24 22451/16
log411/6425 23 42351/64
log10 100,00055 1055100,000
log1/5 255 22 11/5222525
log3 952 3259
loga x
t5log1.05 2.
11.052t52 45log2 16.
24516 y5loga x
Logarithm
For and
. y5loga x means ay5x x.0,
a21, a.0,
EXAMPLE 1
YOUR TURN 1 Write the
equation in logarith-
mic form.
52251/25
EXAMPLE 2
YOUR TURN 2
Evaluate log311/812.
Logarithmic Function
If and then the logarithmic functionof base ais defined by for x.0.
f1x2 5loga x a21,
a.0
2.5 Logarithmic Functions 91 The graphs of the exponential function with and the logarithmic function with are shown in Figure 54. The graphs show that while
Thus, and Also, and In fact,
for any number m, if then Functions related in this way are called inverse functionsof each other. The graphs also show that the domain of the exponential function (the set of real numbers) is the range of the logarithmic function. Also, the range of the exponential function (the set of positive real numbers) is the domain of the logarithmic function. Every logarithmic function is the inverse of some exponential function. This means that we can graph logarithmic functions by rewriting them as exponential functions using the definition of logarithm. The graphs in Figure 54 show a characteristic of a pair of inverse functions: their graphs are mirror images about the line Therefore, since exponential functions go through the point (0, 1), logarithmic functions go through the point (1, 0). Notice that because the exponential function has the x-axis as a horizontal asymptote, the logarithmic function has the y-axis as a vertical asymptote. A more complete discussion of inverse func- tions is given in most standard intermediate algebra and college algebra books.
The graph of is typical of logarithms with bases When the graph is the vertical reflection of the logarithm graph inFigure 54. Because logarithms with bases less than 1 are rarely used, we will not explore them here.
The domain of consists of all In other words, you cannot take the logarithm ofzero or a negative number. This also means that in a function such as
the domain is given by or
Properties of Logarithms The usefulness of logarithmic functions depends in large part on the following properties of logarithms.
x.2.
x22.0, g1x2 5loga1x222,
x.0.
loga x
0,a,1, a.1.
log2 x
y5x.
g1p25m.
f1mf12532p,58 g18253. f122 54 g14252.
g182 5log2 853. f132 52358,
g1x25log2 x
f1x252x
y
x
f x x y x
g x x
FIGURE 54
Properties of Logarithms
Let xand ybe any positive real numbers and rbe any real number. Let abe a positive real number, Then
a.
b.
c.
d.
e.
f. loga ar5r.
loga 150 loga a51 loga xr5r loga x loga
x
y5loga x2loga y loga xy5loga x1loga y
a21.
To prove property (a), let and Then, by the definition of loga- rithm,
Hence,
By a property of exponents, so
Now use the definition of logarithm to write
Since and
loga xy5loga x1loga y.
n5loga y, m5loga x
loga xy5m1n.
am1n5xy.
aman5am1n,
aman5xy.
am5x and an5y.
n5loga y.
m5loga x CAUTION
Proofs of properties (b) and (c) are left for the exercises. Properties (d) and (e) depend on the definition of a logarithm. Property (f ) follows from properties (c) and (d).
Properties of Logarithms
If all the following variable expressions represent positive numbers, then for the statements in (a)– (c) are true.
(a) (b)
(c) TRY YOUR TURN 3
Evaluating Logarithms The invention of logarithms is credited to John Napier (1550 –1617), who first called logarithms “artificial numbers.” Later he joined the Greek words logos(ratio) and arithmos(number) to form the word used today. The development of logarithms was motivated by a need for faster computation. Tables of logarithms and slide rule devices were developed by Napier, Henry Briggs (1561–1631), Edmund Gunter (1581–1626), and others.
For many years logarithms were used primarily to assist in involved calculations. Cur- rent technology has made this use of logarithms obsolete, but logarithmic functions play an important role in many applications of mathematics. Since our number system has base 10, logarithms to base 10 were most convenient for numerical calculations and so base 10 log- arithms were called common logarithms. Common logarithms are still useful in other applications. For simplicity,
.
Most practical applications of logarithms use the number eas base. (Recall that to 7 decimal places, Logarithms to base eare called natural logarithms, and (read “el-en x”). A graph of is given in Figure 55.
NOTE Keep in mind that ln xis a logarithmic function. Therefore, all of the properties of logarithms given previously are valid when ais replaced with eand is replaced with ln.
Although common logarithms may seem more “natural” than logarithms to base e, there are several good reasons for using natural logarithms instead. The most important rea- son is discussed later, in the section on Derivatives of Logarithmic Functions.
A calculator can be used to find both common and natural logarithms. For example, using a calculator and 4 decimal places, we get the following values.
Notice that logarithms of numbers less than 1 are negative when the base is greater than 1.
A look at the graph of or will show why.
Sometimes it is convenient to use logarithms to bases other than 10 or e. For example, some computer science applications use base 2. In such cases, the following theorem is useful for converting from one base to another.
y5ln x y5log2 x
ln 2.3450.8502,
log 2.3450.3692, log 59452.7738, and log 0.00285 22.5528.
loge f1x2 5ln x
loge x is abbreviated ln x e52.7182818.)
log10 x is abbreviated log x loga19x52 5loga 91loga1x52 5loga 915. loga x logax224x
x16 5loga1x224x2 2loga1x162 loga x1loga1x2125loga x1x212
a21, a.0,
EXAMPLE 3
YOUR TURN 3 Write the
expression as a sum,
difference, or product of simpler logarithms.
loga 1x2/y32
f(x)
x f(x) = ln x
0 2 4 6 8
2
–2
FIGURE 55
Change-of-Base Theorem for Logarithms
If xis any positive number and if aand bare positive real numbers, then .
loga x5 logb x logb a
b21, a21, ln 59456.3869, and ln 0.00285 25.8781.
2.5 Logarithmic Functions 93 To prove this result, use the definition of logarithm to write as or (for positive xand positive a, Now take base blogarithms of both sides of this last equation.
Solve for
If the base bis equal to e, then by the change-of-base theorem,
Using ln xfor loge xgives the special case of the theorem using natural logarithms.
loga x5 loge x loge a.
loga x.
loga x5 logb x logb a
loga xr5r loga x logb x5 1loga x2 1logb a2,
logb x5logb aloga x a21).
x5aloga x
x5ay y5loga x
For any positive numbers aand x,
. loga x5 ln x
ln a a21,
The change-of-base theorem for logarithms is useful when graphing on a graphing calculator for a base aother than eor 10. For example, to graph let The change-of-base theorem is also needed when using a calculator to eval- uate a logarithm with a base aother than eor 10.
Evaluating Logarithms Evaluate
SOLUTION As we saw in Example 2, this number is between 2 and 3. Using the second form of the change-of-base theorem for logarithms with and gives
To check, use a calculator to verify that TRY YOUR TURN 4 As mentioned earlier, when using a calculator, do not round off intermediate results. Keep all numbers in the calculator until you have the final answer.
In Example 4, we showed the rounded intermediate values of ln 80 and ln 5, but we used the unrounded quantities when doing the division.
Logarithmic Equations Equations involving logarithms are often solved by using the fact that exponential functions and logarithmic functions are inverses, so a logarithmic equation can be rewritten (with the definition of logarithm) as an exponential equation. In other cases, the properties of logarithms may be useful in simplifying a logarithmic equation.
Solving Logarithmic Equations Solve each equation.
(a) logx 8 2753
5 2.723<80.
log5 805 ln 80
ln 5 < 4.3820
1.6094<2.723.
a55 x580
log5 80.
y5ln x/ln 2.
y5log2 x, y5loga x
EXAMPLE 4