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Exponentialand
Logarithmic Functions
By: Hendrik Pical to Revition
Exponential and Logarithmic Functions
Last Updated: January 30, 2011By Hendry P 2011
With your Graphing Calculatorgraph each of the following
y = 2x
y = 3x
y = 5x
y = 1x
Determine what is happening when the base is changing in each of these graphs.
By Hendry P 2011
y = 2x
x y = 2x y = 3x
-2 1/4 1/9
-1 ½ 1/3
0 1 1
1 2 3
2 4 9
3 8 27
y = 3x
By Hendry P 2011
y = 2x
x y = 5x y = 1x
-2 1/25 1
-1 1/5 1
0 1 1
1 5 1
2 25 1
3 125 1
y = 3x
y = 5x
y = 1x
By Hendry P 2011
y = 2x
y = 3x
y = 5x
y = 1x
Determine where each of the following would lie?
y=10x
y=4x
y = (3/2)x
y = 10x y = 4x
y = (3/2)x
By Hendry P 2011
By Hendry P 2011
f(x) = 2x
By Hendry P 2011
f(x) = 2x-3
By Hendry P 2011
f(x) = 2x+2 - 3
By Hendry P 2011
f(x) = -(2)x-4 – 2
By Hendry P 2011
Compound Interest
ntnrPA 1
You deposit $5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded quarterly?
A = Final amount = unknown
P = Principal = $5000
r = rate of interest = .045
n = number of times compounded per year = 4
t = number of years compounded = 10By Hendry P 2011
Compound Interest
ntnrPA 1
You deposit $5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded quarterly?
A = unknown
P = $5000
r = .045
n = 4
104
4045.015000 A
t = 10
88.7821$A
By Hendry P 2011
Compound Interest
ntnrPA 1
You deposit $5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded quarterly?
A = unknown
P = $5000
r = .045
n = 4
1052
52045.015000 A
t = 10
04.7840$A
weekly?
52
By Hendry P 2011
By Hendry P 2011
With your Graphing Calculatorgraph each of the following
y = 1x
y = (1/2)x
y = (1/3)x
Determine what is happening when the base is changing in each of these graphs.
By Hendry P 2011
y = 2x
Jeff Bivin -- LZHS
x y = (½)x y = (1/3)x
-2 4 9
-1 2 3
0 1 1
1 ½ 1/3
2 ¼ 1/9
3 1/8 1/27
y = 3x
y = 5x
y = 1x
y = (1/3)x
y = (½)x
By Hendry P 2011
f(x) = 2-x = (1/2)x
Jeff Bivin -- LZHS
By Hendry P 2011
f(x) = (½)x-3 - 2 = (2)-x+3 - 2
By Hendry P 2011
By Hendry P 2011
A new Number
!5
1
!4
1
!3
1
!2
1
!1
1
!0
1e
We could use a spreadsheet to determine an approximation.
120
1
24
1
6
1
2
1
1
1
1
1e
0
!1ne
By Hendry P 2011
A new Number
718.20
!1
ne
By Hendry P 2011
y = 2x
x y = 2x y = 3x
-2 ¼ 1/9
-1 ½ 1/3
0 1 1
1 2 3
2 4 9
3 8 27
y = 3x
y = exGraph y = ex
By Hendry P 2011
y = exy = ex+2Graph:
y = ex+2
x + 2 = 0
x = -2
By Hendry P 2011
Compound Interest-continuously
rtPeA
You deposit $5000 into an account that pays 4.5 % interest. What is the balance of the account after 10
years if the interest is compounded continuously?
A = Final amount = unknown
P = Principal = $5000
r = rate of interest = .045
t = number of years compounded = 10
By Hendry P 2011
Compound Interest-continuously
rtPeA
You deposit $5000 into an account that pays 4.5 % interest. What is the balance of the account after 10
years if the interest is compounded continuously?
A = unknown
P = $5000
r = .045
t = 10
10045.05000 eA56.7841$A
By Hendry P 2011
Bacteria Growth
ktney
You have 150 bacteria in a dish. It the constant of growth is 1.567 when t is measured in hours. How
many bacteria will you have in 7 hours?
y = Final amount = unknown
n = initial amount = 150
k = constant of growth = 1.567
t = time = 7
By Hendry P 2011
Bacteria Growth
ktney
You have 150 bacteria in a dish. It the constant of growth is 1.567 when t is measured in hours. How
many bacteria will you have in 7 hours?
y = unknown
n = 150
k = 1.567
t = 7
7567.1150 ey
678.977,706,8ybacteria678.977,706,8
By Hendry P 2011
By Hendry P 2011
y = 2x
x y
-2 1/4
-1 ½
0 1
1 2
2 4
3 8
x y
1/4 -2
½ -1
1 0
2 1
4 2
8 3
x = 2y
By Hendry P 2011
How do we
solve this
exponential
equation
for the variable y?
y = 2x x = 2y
By Hendry P 2011
LOGARITHMS
exponential
logarithmic
b > 0
A > 0
Abm mAb )(log
By Hendry P 2011
2)9(log3 932
exponential
logarithmic
3)125(log5 12553
3log 81
2 8132
5)32(log21 325
21
Abm mAb )(log
yx 2 yx )(log2By Hendry P 2011
u)25(log5
Evaluate
255 u
255 u
2u
2)25(log5
By Hendry P 2011
u)81(log3
Evaluate
813 u
433 u
4u
4)81(log3
By Hendry P 2011
u321
2log
Evaluate
3212 u
5212 u
5u
5log 321
2
522 u
By Hendry P 2011
u)7(log7
Evaluate
77 u
177 u
1u
1)7(log7
By Hendry P 2011
u)1(log8
Evaluate
18 u
088 u
0u
0)1(log8
By Hendry P 2011
unn )(log 5
Evaluate
5nnu
5u
5)(log 5 nn
By Hendry P 2011
y = 2x
x y
-2 1/4
-1 ½
0 1
1 2
2 4
3 8
x y
1/4 -2
½ -1
1 0
2 1
4 2
8 3
x = 2y
y=log2x
By Hendry P 2011
x y = log2x
1/4 -2
½ -1
1 0
2 1
4 2
8 3
y = log2x
y = log3xy = log5x
x = 2y
By Hendry P 2011
x y = log½x
1/4 2
½ 1
1 0
2 -1
4 -2
8 -3
y = log½x
x = (½)y
By Hendry P 2011
Solve for x
log2(x+5) = 424 = x + 5
16 = x + 5
11 = x
By Hendry P 2011
Solve for x
logx(32) = 5
x5 = 32
x5 = 25
x = 2
By Hendry P 2011
Evaluate
log3(25) = u
3u = 25
3u = 52
??????
By Hendry P 2011
By Hendry P 2011
Change of Base Formula
a
xxa
10
10
log
loglog
a
xx
b
ba log
loglog
By Hendry P 2011
Evaluate
log3(25)
= 2.930
3log
25log
10
10
By Hendry P 2011
Evaluate
log5(568)
= 3.941
5log
568log
10
10
By Hendry P 2011
Properties of Logarithms
• Product Property
• Quotient Property
• Power Property
• Property of Equality
By Hendry P 2011
Product Property
nmnm aaa
)(log)(log)(log nmnm bbb multiplication addition
multiplication addition
By Hendry P 2011
Product Property
)4(log)16(log)416(log 222
)2(log)2(log)22(log 22
42
242
24)2(log 62
66
By Hendry P 2011
Quotient Property
nmn
m
aa
a
)(log)(log)(log nm bbnm
b division subtractio
n
division subtraction
By Hendry P 2011
Quotient Property
)4(log)32(loglog 22432
2
)2(log)2(log8log 22
522
25)2(log 32
33
By Hendry P 2011
Power Property
nmnm aa
logb(m p )
logb(mp ) = p•logb(m)
p
By Hendry P 2011
Power Property
)2(log72log 27
2
177
77
By Hendry P 2011
Property of Equality
CAthen
)(log)(log CAif bb
By Hendry P 2011
)(log 235 yx
Expand
)(log)(log 25
35 yx
)(log2)(log3 55 yx
product property
power property
By Hendry P 2011
Expand
)(log)(log 45
355 zyx
)(log)(log)(log 45
35
55 zyx
quotient property
product property
)(log4)(log3)(log5 555 zyx power property
4
35
5logz
yx
By Hendry P 2011
)(log)(log)(log 55
25
75 zyx
)(log)(log 525
75 zyx
Expand
quotient property
product property
)(log5)(log2)(log7 555 zyx power property
52
7
5logzyx
)(log)(log)(log 55
25
75 zyx distributive
property
By Hendry P 2011
zyx 333 log2log6log5
Condense
power property
product property
23
63
53 logloglog zyx
23
653 loglog zyx
2
65
3logz
yxquotient property
By Hendry P 2011
410
21010 logloglog 2
1
zyx
zyx 10101021 log4log2log
Condense
group / factor
product property
4102
1010 logloglog 21
zyx
421010 loglog 2
1
zyx
42
21
10logzyxquotient
property
Power property
4210logzy
x
By Hendry P 2011
4523 loglogloglog wzyx eeee
4253 loglogloglog wyzx eeee
wzyx eeee log4log5log2log3
Condense
re-organizegroup
4253 loglogloglog wyzx eeee
4253 loglog wyzx ee
42
53
logwyzx
e
product property
Power property
quotient property
By Hendry P 2011
By Hendry P 2011
Solve for x
393 xx
122 x
6x
3log93log 33 xx
Property of Equality
By Hendry P 2011
3log93log 33 xx
Solve for x
6xcheck
36log9)6(3log 33
36log918log 33
9log9log 33
checks!By Hendry P 2011
3log93log 33 xx
Solve for x
393 xx
122 x
6x
6By Hendry P 2011
nn 6log2log7log 444
Solve for n
nn 6147
14n
nn 6log)2(7log 44 Condense left side
Property of Equality
By Hendry P 2011
nn 6log2log7log 444
Solve for n
14ncheck
)14(6log214log7log 444
84log12log7log 444
84log)12(7log 44
84log84log 44 checks!
By Hendry P 2011
nn 6log)2(7log 44
nn 6log2log7log 444
Solve for n
nn 6147
14n
14By Hendry P 2011
Solve for x
31log1log 22 xx
)1)(1(23 xx
18 2 x29 xx3
3)1)(1(log2 xxCondense left side
Convert to exponential
form
By Hendry P 2011
Solve for x
31log1log 22 xx
3xcheck 3xcheck
313log13log 22
32log4log 22
312 33
checks!
313log13log 22
34log2log 22
fails
The argument must be positive
By Hendry P 2011
Solve for x
3)1)(1(log2 xx
)1)(1(23 xx
18 2 x29 xx3 3
31log1log 22 xx
By Hendry P 2011
Solve for x
33 22 xx
39 2 xx
60 2 xx
)2)(3(0 xx
23log 23 xx
23 xorx
Convert to exponential
form
By Hendry P 2011
23log 23 xx
Solve for x
3xcheck
checks!
23)3()3(log 23
2339log3 29log3
22
2xcheck
232)2(log 23
2324log3 29log3
22
checks!
By Hendry P 2011
23log 23 xx
Solve for x
33 22 xx
39 2 xx
60 2 xx
)2)(3(0 xx23 xorx 2,3
By Hendry P 2011
Solve for x
)19log()5log()73( x
)19log()5log(7)5log(3 x
)5log(7)19log()5log(3 x
)5log(3
)5log(7)19log( x
943.2x
19log5log 73 x
)5log(3)5log(3
By Hendry P 2011
24 7log5log x
Solve for x
)7log()2()5log()4( x
)5log(4
)7log(2x
605.0x
)5log(4)5log(4
By Hendry P 2011
Solve for x
)9log()5()11log()12( xx
)9log(5)11log(1)11log(2 xx
)11log()9log(5)11log(2 xx
)11log()9log(5)11log(2 x
)9log(5)11log(2)11log(
x
387.0x
xx 512 9log11log
By Hendry P 2011
Solve for x
)5log()1()3log()2( xx
)5log(1)5log()3log(2)3log( xx
)3log(2)5log()5log()3log( xx
)3log(2)5log()5log()3log( x
)5log()3log(
)3log(2)5log(x
12 5log3log xx
151.1xBy Hendry P 2011
Solve for x
)3log()89()7log()23( xx
)3log(8)3log(9)7log(2)7log(3 xx
)3log(8)7log(2)3log(9)7log(3 xx
)3log(8)7log(2)3log(9)7log(3 x )3log(9)7log(3
)3log(8)7log(2x
209.1x
8923 3log7log xx
By Hendry P 2011
Solve for x
)ln()23()15ln( ex
23)15ln( x
x32)15ln(
x3
2)15ln(
x569.1
23ln15ln xe
1
By Hendry P 2011
Solve for x
)ln()65()ln( 37 ex
65)ln( 37 x
x56)ln( 37
x5
6)ln( 37
x 031.1
1
6537 xe
6537 lnln xe
By Hendry P 2011
)2log()13()log( 75 x
Solve for x
)2log(3)2log()log( 75 x
x
)2log(3
)2log()log( 75
x172.0
13275 x
1375 2loglog x
)2log(1)2log(3)log(75 x
By Hendry P 2011
