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Warm-UpWarm-Up
Solve for x in each equation.
a) 3x = b) log2x = –4
c) 5x = 30
Logarithmic, Exponential, and Other Transcendental Functions 20145
Copyright © Cengage Learning. All rights reserved.
If you aren't in over your head, how do you know how tall you are? T. S. Eliot
Chapter 5Chapter 5
Transcendental Functions:Transcendental Functions:Bases Other than e
Day 1: All rules and derivative examples.Day 1: All rules and derivative examples.
Day 2: Integration examples and applications.Day 2: Integration examples and applications.
http://www.youtube.com/watch?v=SNZgbj3UaRE
Recall that the natural log, ln x, is a logarithm with base e. We can differentiate ln x simply with the following rule:
Bases Other than eBases Other than e
xx eedx
d
In addition, we know that the natural exponential function, ex, can easily be differentiated:
x
xdx
d 1ln
ueedx
d uu u
uu
dx
d ln
By applying the chain rule, we get:
and
But, what do we do if we have a logarithm or exponential function that has a base other than e? To determine these derivatives, we need to use a very useful logarithmic operation called the change of change of base formulabase formula. Recall from precalculus:
Bases Other than eBases Other than e
a
b
a
bb
e
ea ln
ln
log
loglog
This formula allows us to transform any logarithm into a quotient of logarithms with any base that we choose, including the natural logarithm. More specifically:
a
bb
c
ca log
loglog
a
x
dx
d
ln
ln
Now let’s try to find the derivative of a logarithm with a base other than e:
Bases Other than eBases Other than e
xdx
dalog
This is a constant
xa
1
ln
1 x
dx
d
aln
ln
1
Bases Other than eBases Other than e
To determine the derivative of a natural exponential function with a base other than e, we need to note:
axedx
d ln xadx
d
ae ax lnln
axax eeax lnln
xaaln
l nu a axu ln
ueedx
d uu
So, how do we integrate an exponential function with a base other than e? Once again, we use the alternate form of the exponential function of ax:
In terms of integration, at this point, we cannot find the integral of ln x, but we can integrate ex. Recall:
Bases Other than eBases Other than e
Cedxe xx
Cea
u ln
1 duea
u
ln
1
dxe ax ln dxa x
Cea
ax ln
ln
1
axax eeax lnln
1
lnxa C
a
lnd u a d xaxu ln
Caa
dua uu ln
1.6
The rules to differentiating with bases other than e
Rules for derivatives of bases Rules for derivatives of bases Other than eOther than e
xa
xdx
da
1
ln
1log.1
Caa
dxa xx ln
1.5
u
u
au
dx
da
ln
1log.2
xx aaadx
dln.3 uaaa
dx
d uu ln.4
The rules to integrating with bases other than e
)23(77ln 2123
xxxx
Differentiate:
ExamplesExamples
x
x
xy
cos
sin
4ln
11
4ln
12
xxy coslog.2 24 xx c o slo glo g 4
24
123
7.3 xxy
xxy 8.4 3
uy u 77ln
xxdx
du23 2 123 xxu
3 22 1(3 2 )(7 ) ln 7x xx x
8ln883 32 xx xxy
xxy 5log.1 27
)5)(7(ln
522 xx
x
xx
xy
5
52
7ln
12
xx
tan4ln
1
4ln
2
4ln
tan
4ln
2 x
x
xx c o slo glo g2 44
xx
tan2
4ln
1
u
u
au
dx
da
ln
1log uaaa
dx
d uu ln
Differentiate:
Practice ProblemsPractice Problems
352 42log.1 xy
)42(2ln
305
4
x
x
42
10
2ln
13
5
4
x
xy
41 0 xu 52 4u x 42log3 52 x
u
u
au
dx
da
ln
1log
)2(2ln
155
4
x
x
Differentiate:
Practice ProblemsPractice Problems
27
2log.2
x
xy
xxy
1
7ln
12
2
1
7ln
1
u
u
au
dx
da
ln
1log
)2)(7(ln
42
)2)(7(ln
xx
x
xx
x
xx )7(ln
2
)2)(7(ln
1
)2)(7(ln
4
xx
x
277 log2log xx xx 77 log22log
)2)(7(ln
42 xx
x
Differentiate:
Practice ProblemsPractice Problems
1. xy
2 sin2. 10x xy
53. 9xy x
xxxx c o s21 01 0ln s i n2
uaaadx
d uu ln
1 0ln1 0c o s2 s i n2 xxxx
9ln995 54 xx xx 54 99ln95 xx xx
xxu c o s2 xxu s in2
lnxx ln
lnx xda a a
dx
5.5 Homework Day 1 AB5.5 Homework Day 1 AB Page 366 1, 7, 19, 21, 27, 37-51 odd, 57
5.5 Homework Day 1 BC5.5 Homework Day 1 BC Page 366 21-55 odds
18
Chapter 5Chapter 5
Transcendental Functions:Transcendental Functions:Bases Other than e
Day 2: Integration examples and Applications.Day 2: Integration examples and Applications.
Find an equation of the tangent line to the graph of .
HWQHWQ
10log 2 5,1y x at
11 5
5 ln10y x
Occasionally, an integrand involves an exponential function to a base other than e. When this occurs, there are two options:(1) use substitution, and then integrate, or (2) integrate directly, using the integration formula
Day 2: IntegrationDay 2: Integration
Example – Integrating an Exponential Function to Another Base
Find ∫2xdx.
Solution:
∫2xdx = + C
Integrate:
dxx 73.5
ExamplesExamples
Cu 55ln
1
2
1
dxx x2
5.6 d xxd u 22xu
duu 52
1C
x
5ln2
52
Cu 33ln
1
7
1
d xd u 7xu 7
duu 37
1 Cx
3ln7
37
Integrate:
Practice ProblemsPractice Problems
dxx 65.7
dxx x 2)3(7)3(.8
Cx
7ln2
72)3(
Cx 2)3(77ln
1
2
1
d xxd u 322)3( xu
duu 56
1 Cx
5ln6
56Cu 5
5ln
1
6
1
d xd u 6xu 6
duu 72
1
Applications of Exponential Functions
Applications of Exponential Applications of Exponential FunctionsFunctions
Suppose P dollars is deposited in an account at an annual interest rate r (in decimal form). If interest accumulates in the account, what is the balance in the account at the end of 1 year? The answer depends on the number of times n the interest is compounded according to the formula
A = P
Applications of Exponential Applications of Exponential FunctionsFunctions
For instance, the result for a deposit of $1000 at 8% interest compounded n times a year is shown in the table.
Applications of Exponential Applications of Exponential FunctionsFunctions
As n increases, the balance A approaches a limit. To develop this limit, use the following theorem.
Applications of Exponential Applications of Exponential FunctionsFunctions
To test the reasonableness of this theorem, try evaluating
it for several values of x, as shown in the table.
Applications of Exponential Applications of Exponential FunctionsFunctions
Now, let’s take another look at the formula for the balance A in an account in which the interest is compounded n times per year. By taking the limit as n approaches infinity, you obtain
Applications of Exponential Applications of Exponential FunctionsFunctions
This limit produces the balance after 1 year of continuous compounding. So, for a deposit of $1000 at 8% interest compounded continuously, the balance at the end of 1 year would be
A = 1000e0.08
≈ $1083.29.
Applications of Exponential Applications of Exponential FunctionsFunctions
Example 6 – Comparing Continuous, Quarterly, and Example 6 – Comparing Continuous, Quarterly, and Monthly CompoundingMonthly Compounding
A deposit of $2500 is made in an account that pays an annual interest rate of 5%. Find the balance in the account at the end of 5 years if the interest is compounded (a) quarterly, (b) monthly, and (c) continuously.
Solution:
Example 6 – Example 6 – SolutionSolution
cont’d
Example 6 – Example 6 – SolutionSolution Figure 5.26 shows how the balance increases over the five-year period. Notice that the scale used in the figure does not graphically distinguish among the three types of exponential growth in (a), (b), and (c).
Figure 5.26
cont’d
5.5 Homework Day 25.5 Homework Day 2 Page 366 59-71 odds, 83, 85