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CHAPTER 2 : EXPONENTS & LOGARITHMS
2.1 EXPONENT2.2 LOGARITHMS2.3 EXPONENT & LOGARITHMS EQUATION
INTRODUCTION
Why study exponential & logarithmic functions?
They are very important in many technical areas, such as business, finance, nuclear technology, acoustics, electronics & astronomy.
Many of the applications will involve growth (INCREASING) or decay (DECREASING).
There are many things that grow exponentially, for example population, compound interest & charge in capacitor.
We can also have exponentially decay for example radioactive decay.
Logarithm is a method of reducing long multiplications into much simpler additions (and reducing divisions into subtractions).
xxf 2)( Example:Graph the function Solution;Produce the table values of x from -2 to 3.
If a is any real number & n is a positive integer, then the n – th power of a is ;
2.1 EXPONENT
aaaan ....
Definition
x -2 -1 0 1 2 3
f(x) 0.25
base
Exponent (index / power)
2.1 EXPONENT
Law Example Try
am × an = am+n x3x7 = x3+7 = x10 x2x-5 =
am ÷ an =am-n
(am)n = amn (43)2 = 43(2)=46 (55)2 =
(ab)n = anbn (2b)3 = 23b3= 8b3 (3xy)4 =
Law of exponents
2646
4 kk
k
k 2
5
h
h
2.1 EXPONENT
Law Example Try
Law of exponents
n
nn
b
a
b
a
n
nn
a
b
b
a
nn
aa
1
81
16
3
2
3
24
44
2
4
w
8
1
2
12
33 23
4
25
2
5
5
22
22
3
3
4
2.1 EXPONENT
Radical, Rational, - ve & Zero exponent Radical : √ “ the positive square root of “
abba nn means
a ≥ 0, b ≥ 0
n – th root, n any +ve
integer
Rational exponent : n mmnnm
aaa
10 a
m & n are integers, n > 0
Negative exponent : nn
aa
1
Zero exponent :
PRACTICE 1
1. Evaluate the expression.
2/34/3 42 f)
53 44 a)
2/199 c)
323 b)
32
3
1 d)
2/14325 a)
yx
8
54
3
33 e)
52 34 b) xx
3
4
3
6 c)
a
a
3/52/3 d) yy
23224 e) zyx
0
f)yx
yxyx
2. Simplify the expression.
Logarithm function with base a, denoted by loga is defined by;
2.2 LOGARITHMS
813
8238log
12553125log
4
32
35
Definition
base
Exponent (index / power)
form lexponentiaform logaritmic
log yaxy xa
Example:
Equivalent form
b
x
b
x
b
xx
a
ab ln
ln
log
log
log
loglog
10
10
Common logarithm : Logarithm with base 10, denoted by,
Natural logarithm : Logarithm with base e, denoted by
Base conversion :
Type of Log
yy 10loglog
yy elogln
Any base Base
10
Base e
2.2 LOGARITHMS
Example 1
1. Rewrite each function below in exponential or logarithm form.
a) 72 = 49 b) Log2128 = 7c) 5-2 = 1/25d) Logb1=0
2. Determine the value of log27 and log3 12.8074.2
2log
7log7log
10
102
Logarithms Example
loga xy = loga x + loga y log 45x = log 45 + log 4x
loga (x/y) = loga x − loga y ln 8 – ln 2 = ln (8/2) = ln 4
loga (xn) = n loga x log 53 = 3log 5
loga a = 1 log33 = 1
loga 1 = 0 ln 1 = 0
Law of logarithms
2.2 LOGARITHMS
Example 2
1. Use the property of logarithms to rewrite each of the
following:a) ln 18 = ln (2.3.3) =b) log 5 + log 2 =c) log (3/5) =d) log 8x2 – log 2x = e) Log 1003.4 = log (102)3.4 =
2. Simplify & determine the value of ; 2log 5 + 3log 4 – 4log 2
PRACTICE 2
If log 2 = 0.3010, log 3 = 0.4771 and log 5 = 0.6990, determine each of the following without using a calculator:a) log 6 = log 2x3 = log 2 + log 3
= 0.3010 + 0.4771 = 0.7781b) log 81
c) log 1.5
d) log √5
e) log 50
Exponential Equation The variable occurs in
the exponent. E.g. 2x = 7 To solve:
1) Use the properties of exp.
2) Rewrite in equivalent form.
3) Solve the resulting equation.
Logarithmic Equation A logarithm of the
variable occurs. E.g. log2 (x+2) = 5 To solve:
1) Use the properties of log.
2) Rewrite in equivalent form.
3) Solve the resulting equation.
2.3 EXPONENTIAL & LOGARITHS EQUATION
Example 3
calculator a use, for Solve 43ln
81ln
exponent) the down (bring 3Law 81ln3ln
side each of ln Take 81ln3ln
xx
x
x
Solve each of the following;a) 3x = 81
b) 52x+1 = 254x-1
2
1
for Solve 2812
4Law and 3Law ly App5log285log12
side each of log Take 5log5log
55
55
528
512
5
14212
x
xxx
xx
xx
xx
for Solve 0.4582x
ln ofProperty 0.91632x
side each of ln Take5.2lnln
8by Divide8
20
208
2
2
2
x
e
e
e
x
x
x
Example 3 Solve each of the following;
c) 8e2x = 20
calculator Use 1096.6
form Equivalent 7
ex
Solve each of the following;a) ln x = 7
b) log2 (x+2) = 5
c) Log2(25 – x) = 3
30
for Solve 232
form lExponentia 22
52log5
2
xx
x
x
Example 4
Example 4
2by Divide5000
form lExponentia102
3by Divide42log
4 Subtract122log3
4
x
x
x
x
Solve each of the following;d) 4 + 3log 2x = 16
e) C
f) c
3lnln2ln2 x
124ln3ln2 xx
PRACTICE 3
Solve each of the following.
452log f) 2 x
8 a) 4.0 te
312 c) 4.0 te
65b) 2 te
3log d) 2 x
32loglog g) 22 xx
x227log e) 3
11log1log h) 33 xx