Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.
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Transcript of Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.
![Page 1: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/1.jpg)
Warm-UpWarm-Up
Solve for x in each equation.
a) 3x = b) log2x = –4
c) 5x = 30
![Page 2: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/2.jpg)
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
![Page 3: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/3.jpg)
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.
![Page 4: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/4.jpg)
http://www.youtube.com/watch?v=SNZgbj3UaRE
![Page 5: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/5.jpg)
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
![Page 6: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/6.jpg)
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
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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
![Page 8: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/8.jpg)
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
![Page 9: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/9.jpg)
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
![Page 10: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/10.jpg)
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
![Page 11: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/11.jpg)
)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
![Page 12: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/12.jpg)
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
![Page 13: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/13.jpg)
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
![Page 14: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/14.jpg)
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
![Page 15: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/15.jpg)
5.5 Homework Day 1 AB5.5 Homework Day 1 AB Page 366 1, 7, 19, 21, 27, 37-51 odd, 57
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5.5 Homework Day 1 BC5.5 Homework Day 1 BC Page 366 21-55 odds
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18
Chapter 5Chapter 5
Transcendental Functions:Transcendental Functions:Bases Other than e
Day 2: Integration examples and Applications.Day 2: Integration examples and Applications.
![Page 18: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/18.jpg)
Find an equation of the tangent line to the graph of .
HWQHWQ
10log 2 5,1y x at
11 5
5 ln10y x
![Page 19: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/19.jpg)
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
![Page 20: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/20.jpg)
Example – Integrating an Exponential Function to Another Base
Find ∫2xdx.
Solution:
∫2xdx = + C
![Page 21: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/21.jpg)
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
![Page 22: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/22.jpg)
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
![Page 23: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/23.jpg)
Applications of Exponential Functions
![Page 24: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/24.jpg)
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
![Page 25: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/25.jpg)
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.
![Page 26: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/26.jpg)
Applications of Exponential Applications of Exponential FunctionsFunctions
As n increases, the balance A approaches a limit. To develop this limit, use the following theorem.
![Page 27: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/27.jpg)
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.
![Page 28: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/28.jpg)
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
![Page 29: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/29.jpg)
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.
![Page 30: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/30.jpg)
Applications of Exponential Applications of Exponential FunctionsFunctions
![Page 31: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/31.jpg)
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:
![Page 32: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/32.jpg)
Example 6 – Example 6 – SolutionSolution
cont’d
![Page 33: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/33.jpg)
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
![Page 34: Warm-Up Solve for x in each equation. a) 3 x = b) log 2 x = –4 c) 5 x = 30.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d755503460f94a55187/html5/thumbnails/34.jpg)
5.5 Homework Day 25.5 Homework Day 2 Page 366 59-71 odds, 83, 85