Alg 3 Functions 1
Algebra 3 Assignment Sheet
Logs (1) Assignment # 1 – Exponential Equations (2) Assignment # 2 – Logarithms (3) Assignment # 3 – Laws of Logarithms (4) Assignment # 4 – Calculator Problems (5) Assignment # 5 – Review Worksheet (6) TEST
Alg 3 Functions 2
Alg 3 Functions 3
5.1 Exponents
EXPONENTIAL EQUATIONS
I Review
x yb b = y
xb
x
y
b
b
0b
xb x xb c
3 3b b 2
3b
3
2
b
b
II x yIf b b , then x y
x xIf b a , then b a
If x = y, then x y b b
Ex. x 2 53 3 Ex.
x 42 2
3 3
x+2 = 5 x = 4 x = 3
2x 1 x 16
2x 1
x
1) 2 32 2) 2 2
13) 8 64 4) 64
2
Alg 3 Functions 4
2 2x 1 x x x 5
x x x x x
15) 27 6) 8 4
9
17) 3 3 3 8) 5 4 4 96
729
x x x x 1
9) 3 3 3 3 54
10)
1
3 2
2
2 3
-
- -
11) 2
14
8x + 1 x 3
= 16
12)
3x3x 24 9 4 + 8 = 0
Alg 3 Functions 5 III Sketching
xy 2 Notes
xy 2
How could you sketch its inverse?
-3
-2
-1
0
1
2
3
x y
-3
-2
-1
0
1
2
3
x y
Alg 3 Functions 6
Graph xy = 1
Graph xy = 2
Graph its inverse on the same graph
* Remember how to recognize quadratic form equations
Alg 3 Functions 7
Algebra 3 Assignment # 1
Exponential Equations (1) Solve for x please.
(a) x1 x 8 4 (b) 5
2513 x
(c) 7 x 1 x2 8 2 (d)
1 x2x x 2227 9
(e) 41
x 11 x2 16 8 (f) 16 1 x 34 2
(g) 8
27 x 23
(h) 13 2 x2 0 48 2x
(i) 10 27 x2 0 9 27x (j) 10 16x 0 16 4x
(2) Sketch a graph of each of the following on the same graph.
(a) x3 y (b)
x3 y
Alg 3 Functions 8
Algebra 3 Assignment # 1
Answers
(1) (a) 2 (b) 5
(c) 20 (d) 1 , 3
(e) 910
(f) 3
(g) 94 (h) 4
(i) 0 , 32 (j)
2
3
21 ,
Alg 3 Functions 9
EXPONENTIAL EQUATIONS EXTRA (1) Evaluate each of the following numbers please.
(a)
3 12 2
9 49 (c) 2 2 2
2 4
(b)
2 2
1
2 + 2
2 (d)
1
3 2
2 2 3
(2) Solve each of the following equations please.
(a)
23
x = 25 (e)
32 5
x + 7 = 8
(b)
35 1
x = 8
(f)
32 4
x 6x + 9 = 27
(c) 2x + 3 x + 4
8 16 (g) 2x x
8 6 8 + 8 = 0
(d) 2x + 1 x 31
48 = 16 (h)
3x3x 24 9 4 + 8 = 0
(3) Sketch a graph of each of the following on the same set of axes.
x1
y = 2
and
y1
x = 2
Alg 3 Functions 10
EXPONENTIAL EQUATIONS EXTRA
Answers
(1) (a) 277
(c) 1
(b) 172
(d) 6
(2) (a) 125 (e) 5 (b) 32 (f) 12 , –6
(c) 72
(g) 1 23 3
,
(d) 138
(h) 0 , 1
Alg 3 Functions 11 5.2 LOGS
I xy 3 inverse? We call this exponent a logarithm
( “log” for short)
Def: y
b b
For x 0, and b 0, and b 1, the logarithmic function with base b is denoted
where if and only if f(x) = log x, y = log x x = b
Logarithms are exponents and follow exponent rules. LOGARITHM FORM EXPONENTIAL FORM
16
1
13
27
2
4
2
3
Log form Exponential form
log 8 = 3
log = 2
log = 0
log =
Exponential form Log form
33 27
2 1
525
32 8
x4 1
base
power number
blog n p pb n
base
power number
Alg 3 Functions 12 RULES
4
1
x
b
b
1) n > 0, p must be real ) log b = x
2) b 1 5) log = 0
1 2 b b 1 2blog x
3) If log x = log x , then x = x 6) b = x
EXAMPLES
1. 5
log 25 =
2. 3
log 3 =
3. b
log 16 = 2
4. b
log b =
5. 6
log x = 1
6. Find the inverse of 2
log 2x + 4
Alg 3 Functions 13 MORE EXAMPLES
3 8
2 27
5
log 81 x log 1 c
log 4 y log 9 d
log 25 z 9 3
8 125
6
1 log log x
2
2log 8 p log x
3
log 6 a
Alg 3 Functions 14 Examples
1) 23
8log =
27 2)
b
1 3log =
27 2
3) 3 14 27
1
81log (log ) x 4) 10.001log y
Alg 3 Functions 15
Compound interest arises when interest is added to the principal, so that from that moment on, the
interest that has been added also itself earns interest. This addition of interest to the principal is called
compounding. A bank account, for example, may have its interest compounded every year: in this case,
an account with $1000 initial principal and 20% interest per year would have a balance of $1200 at the
end of the first year, $1440 at the end of the second year, and so on.
20% growth on a $1000 investment
P = principal amount (the initial amount you borrow or deposit)
r = annual rate of interest (as a decimal)
t = number of years the amount is deposited or borrowed for.
A = amount of money accumulated after n years, including interest.
n = number of times the interest is compounded per year
Wikipedia
Alg 3 Functions 16
http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_Exponents_e.xml
Natural Log of (e) = ln (e)
A. A valuable number for calculus and compound interest
n
11 +
n
If you took this to many places on the calculator you would see it approaches the number 2.71828…
This number is designated as “e”.
It is used often with logs and has its own designation.
If e is the base , then the exponential function is xy = e , and
the log function is expressed as y = ln e . It is assumed the base is e when using ln instead of log.
B. Properties of “e”
C. Examples
1) 4 ln( e) = x 2) 3ln e2 = x
Alg 3 Functions 17
Algebra 3 Assignment # 2 Definition of the logarithm
Solve for x please.
(1) x 64 log4 *(10)
3 2 = x5ln e
(2) 2 x log6 (11) 2
251
xlog
(3) 2 9 logx (12) x 216 log36
(4) 2 xlog3 (13)
23
4 x log
(5) x 125 log25 (14) x 24 log8
(6) 32
8 x log (15)
21
x 6 log
(7) x 81log27 *(16)
2ln ex = 5
(8) x 7 log7 *(17)
21
24 x loglog
(9) 43
16 x log *(18)
41
x16 9 loglog
Alg 3 Functions 18
Algebra 3 Assignment # 2
Answers
(1) 3 (10) 103
(2) 36 (11) 5
(3) 3 (12) 23
(4) 91 (13)
81
(5) 23 (14)
65
(6) 4 (15) 361
(7) 34 (16) 25
(8) 21 (17) 4
(9) 81 (18) 3
Alg 3 Functions 19
5.3 LAWS OF LOGS
Properties of Logs I LAWS II Equations
b b b
b b b
pb b
log mn log m log n
mlog log m log n
n
log m plog m
1) 3 3 3
log 6 log 2 log 3 5) 2
5 5log (2n + 20) = log (32 5n )
2) 5 5 5
xlog log x
5log 5
3) 4 4
3l g 4 43logo 6) 3 3x = 4 , then x = 4
Now go backwards III Examples
7) 6 6 6log 48 log w log 4 4) 2 2 2log 3 log 7 log x
b b
nb
a b
If log m log n then m n
If log m n then
log =log
(log number)
b m
If a = b, then x = x
Alg 3 Functions 20
8) 10 10
1log m log 81
2 9) 7 7 7
1 1log m log 64 log 121
3 2
10) 2 2log ( 2) 1 log 2y y 11) 2
8log 4 = x
12) 10 10log y log y 21 2
Uh, oh, what is this????
13) 3log 53 = x 14) 16log 94
15) ln 3e = 16)
2ln e =
Alg 3 Functions 21
What happens if you have a problem where the bases are not the same?
Example: 2
log 3 = x
Change of Base Rule:
Alg 3 Functions 22
Algebra 3 Assignment # 7 Properties of Logarithms
(1) Evaluate each of the following please.
(a) 7log 67 (b) 25log 36
5
(c) 8 4log 27 log 254 + 8 (d)
2ln 8 3ln 4e
(2) Solve for x please.
(a) 3 3
2x 1 3x 6log log (b) 2
10x + 9x = 1log
(c) 5 5
x = 4 3log log (d) 1 19 9 92 3
x = 144 8log log log
(e) 3 3 3
7 + x 2 = 6xlog log log (f) 15 + 14 105 = xln ln ln ln
(g) 10 10 7
x 1 + x + 2 = 7log log log (h) 3 3 3
x + 3 + x 3 = 16log log log
(i) 8 8 8
x + 1 x = 6x + 2log log log (j) 3 3 3
x + 3 + 4x 1 = 12log log log
(k) 2 2
8 8 3x x 2x 5 = log log (l) 4 9log logx 9 4 = 125 8 3
Alg 3 Functions 23
Algebra 3 Assignment # 7
Answers (1) (a) 6 (b) 6 (c) 134 (d) 1
(2) (a) 7 (b) 10 , 1
(c) 9 (d) 6 (e) 14 (f) 2 (g) 3 (h) 5
(i) 13
(j) 1
(k) 4 , 5 (l) 23
Alg 3 Functions 24
MORE LOG EQUATIONS
NOTE:
1
2 24 2 , 2 4 ,
11
2 329 3 , 3 9 , 3 27
I Solve for x please
1) 49log 27 x 2)
21 1
3 3
log 12 log 20 9 1x x
3) 1
3
log 2x 4) 2
4 42 log 5log 12 0x x
5) 9log 49 x 6) 2log 6
4 b
7) 4log x8 8 8)
log 52x 2125 16
Alg 3 Functions 25
Algebra 3 Assignment # 8 Properties of Logarithms 2
Solve for x please.
(1) 2 1
4 4 2x 1 5x 11 = log log
(2) 2
6 6 63x 5 x 1 = x 1log log log
(3) 24x + 1 + x + x = 19x 9ln ln ln
(4) 2 2 2
8 8 33x 7 x x 1 = log log
(5) 2x + 4 + 3x 4 = 17x 18ln ln ln
(6) 92ln 3 log 25
2 2x + 1 + x 5 = e 3log log
* (7) 3 9
x 5 = x + 7log log ( change of base)
(8) 2
8 83 x x 2 = 0log log
(9) 2
4 42 x + x = 0log 5log
(10) 22 x = 3ln
Alg 3 Functions 26
Algebra 3 Assignment # 8 Answers
(1) 3 , 7
(2) 2 , 3 , reject 5
(3) 34
1 , , reject 3
(4) 3 , (reject 1)
(5) 13
2 , reject 1 ,
(6) 7 , (reject 3)
(7) 9 , (reject 2)
(8) 14
8 ,
(9) 132
, 1
(10) Ø
Alg 3 Functions 27
Using calculators
1) x3 8 2) 2log 6 x
3) 3y y 36 8 4)
4x 7 2x 33 4
5) 1) b b
If log 3 = .4771 and log 2 = .3010
b bfind a) log 12 b) log 1.5
Alg 3 Functions 28
6) x
.03 5
Alg 3 Functions 29
Algebra 3 Assignment # 9 (1) Use a calculator to solve each of the following correct to 4 decimal places please.
(a) x5 = 20 (b)
3x + 1 1 x4 = 9
(c) 3
18 = xlog (d) 7
x = 1.432log
(e) x = 1.432ln (f) 3xlog
5 = 11
(g) x0.3 > 7 (h)
22 x 5 x 3 = 0ln ln
(2) Let 10
2 = plog and 10
3 = qlog . Evaluate each of the following in terms of p and q.
(a) 10
6log (b) 10
72log
(c) 10 5
3 3
16log (d)
1090log
(e) 10
0.5log (f) 10
5log
(3) Simplify the following expression please.
4 49 25
125 32 7 log log log
(4) The magnitude of an earthquake is measured using the Richter scale;
4.4
2 EM =
3 10log ,
Where M is the magnitude of the earthquake, and E is the seismic energy released by the
earthquake (in joules). The 1989 San Francisco earthquake released approximately 151.12 x 10
joules. Calculate the magnitude of the earthquake using the Richter scale. How much energy would be released (in joules) by an earthquake which measures 8.3 on the Richter scale?
Alg 3 Functions 30
Algebra 3 Assignment # 9 Answers
(1) (a) 1.8614 (b) 0.1276 (c) 2.6309 (d) 16.2248 (e) 4.1871 (f) 5.1388
(g) x < 1.6162 (h) 0.6065 , 20.0855 (2) (a) p + q (b) 3p + 2q
(c) 3 42 5
q p (d) 2q + 1
(e) p (f) 1 p
(3) 158
(4) 7.1 , 167.079 x 10 joules
Alg 3 Functions 31
Algebra 3 Assignment # 10 ─ Review Worksheet
(1) Solve for x please.
(a) 2–x 2x+19 = 27 (i) 9log 4
9 = x
(b) 2x–5 x+18 = 16 (j) 9 3log 4 log 4
3 – 9 = x
(c) 1–x2x 1
164 8 = (k)
28 83 log (x) – 2 log (x) – 1 = 0
(d) 3
8log 4 = x (l) 5 5 log 2x 3 = log 1– x
(e) 14
1log x = –
2 (m) 2 2log x 1 + log 3x–1 = 5
(f) x4
log 16 = –3
(n) 2 2 2log x – 3 – log x+1 = log 8
(g) xlog .125 = 3 (o) 7 7 7 7log x 1 + log x + log 2x 1 = log 30
(h) 3 8log log x = –1 (p) 24 4
log 6log 2 log x4 + 4 = 8
(2) Use a calculator to solve for x. Express answers correct to 3 decimal places.
(a) x3 = 8 (b)
3x–2 1–x2 = 5
(c) 3log 2 = x (d) 3
x 4 x = 0ln ln
Alg 3 Functions 32
Algebra 3 Assignment # 10 ─ Review Worksheet Answers
(1) (a) 8
1 (i) 4
(b) 2
19 (j) –14
(c) –3 (k) 8 , 2
1
(d) 9
2 (l)
3
2–
(e) 2 (m) 3
(f) 8
1 (n)
(g) 2
1 (o) 2
(h) 2 (p) 4 (2) (a) 1.893 (b) 0.812 (c) 0.631 (d) 1 , 7.389 , 0.135
Alg 3 Functions 33 Logs & Twigs
(1) 3 3 3
x + 3 + 4x 1 = 12log log log 154
1 , reject
(2) 2 2
8 8 3x x 2x 5 = log log 4 , 5
(3) 2ln 4x + 1 + ln x + x = ln 19x 9 3
41 , , reject 3
(4) 2
6 6 63x 5 x 1 = x 1log log log 2 , 3 , reject 5
(5) 2 1
4 4 2x 1 5x 11 = log log 3 , 7
(6) 2 2 2
8 8 33x 7 x x 1 = log log 3 , (reject 1)
(7) 2x + 4 + 3x + 4 = 17x 18ln ln ln 1
32 , reject 1 ,
(8) 92ln 3 log 25
2 2x + 1 + x 5 = e 3log log 7 , (reject 3)
(9) 3 9
x 5 = x + 7log log 9 , (reject 2)
