indices, surds and logarithm - lydiavalensia - … grade x sma negeri 2 sekayu ___lydia valensia...
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1
Lydia Valensia
X Grade
Mathematics is the way you think
INDICES, SURDS AND
LOGARITHM
www.freedomroad.org.uk
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GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
Hello, my students….
Today, we are going to learn about indices, surds and logarithms.
You know, actually indices, surds and logarithms are closely related.They
are most of the time,studied together. So now, lets begin…
STANDARD COMPETENCE:
1. Solving problems related to indices, surds and logarithms BASIC COMPETENCE:
Using laws of indices, surds and logarithms.
Doing the algebraic manipulation in computation related to indices, surds and logarithms.
(Integrated with E- SETS )
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GRADE X
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1.1 INDICES
A. PROPERTIES OF EXPONENT
Have you ever bought eggs ? How many eggs that you
get, if you buy 8 eggs each day for 8 days?
Ok. This is easy question, isn’t it?
So, what is the result? Yups, that is right 64.
Do the result 64 comes from 8 x 8 ?. We can write 8 x 8 = 82.
It means that we have used indices.
What is indices ?
Now, Lets discuss it.
Indices or powers are also called exponent
The exponent of a number says how
many times to use the number in
multiplication.
In example 82 = 8 x 8 = 64. How to
read 82? Don’t miss it !
Need more example? Here they are.
35 = 3 x 3 x 3 x 3 x 3
In words : 35could be called “ 3 to the fifth power”, “3 to the power 5”.
And exponents make it easier to write and use many multiplications.
Example :
Recurring Number Magic
Activity: You write down the following 8 digit number on a piece of paper:
1 2 3 4 5 6 7 9
Then ask a friend to circle one of the digits. Say that they circle number 7.
You then ask your friend to multiply the 8 digit number by 63, and magically the result ends up being:
1 2 3 4 5 6 7 9 x 6 3 7 7 7 7 7 7 7 7 7
with the answer as a row of the chosen number 7.
How about if your friends circle number 3? Ask them to multiply by 27 and the result is 333333333.
What is the secret ?
In words: 82 could
be called ―8 to the second power‖, ―8 to the power 2‖ or simply ―8 squared‖.
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GRADE X
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116 is easier to write and read than 11 x 11 x 11x 11 x 11x 11
You can multiply any number by itself as many times as you want using this
notation.
So, in general:
THE KEY OF EXPONENT
The "Laws of Exponents" (also called "Rules of Exponents"), all come from
three following ideas:
1. The exponent of a number says to multiply the number by
itself so many times
2. The opposite of multiplying is dividing, so a negative
exponent means divide
3. A fractional exponent like 1/n means to
take the nth root:
If you understand those, trust me that you are able to continue
next journey of exponent.
All the laws below are based on those ideas.
Laws of Exponents
Here are the Laws (explanations follow):
Law Example
x1 = x 61 = 6
x0 = 1 70 = 1
x-1 = 1/x 4-1 = ¼
xmxn = xm+n x2x3 = x2+3 = x5
xm/xn = xm-n x4/x2 = x4-2 = x2
(xm)n = xmn (x2)3 = x2×3 = x6
(xy)n = xnyn (xy)3 = x3y3
(x/y)n = xn/yn (x/y)2 = x2 / y2
x-n = 1/xn x-3 = 1/x3
Puzzle Math Aptitude Test Which of the following sentences is correct? Nine and five are thirteen. or Nine and five is thirteen. Solution: Neither is correct ,
9 + 5 = 14.
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GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
Laws Explained
The first three laws above (x1 = x, x0 = 1 and x-1 = 1/x) are just part
of the natural sequence of exponents.
Have a look at this example:
Simplify and write down in positive exponents
1. 4
422
2
2
2
75 1.
yyy
y
y
y
yy
3
31
)3(6)3(2
3362
313232
323
2
1
2
.2
2.2
2.22
.2.2
a
a
a
aa
aaa
a
3. Evaluate the following without using calculator :
(a) 33
4
88
25
9
5
3
5
3
5
3
125
27
125
272
2
3
32
3
32
33
2
b
If you find it hard to remember all of these rules, then remember this:
“ You can always work them out if you understand the three that
have explained”.
Ups, what will happen if x (variable) = 0
Positive Exponent (n>0) 0n = 0
Negative Exponent
(n<0) Undefined! (Because dividing by 0)
Exponent = 0 Ummm ... (see column did you
know)
Did You Know
The Strange case
of 00
There are two
different arguments
for the correct
value. 00 (zero to the
power zero) could be
1, or possibly 0, so
some people say it is
really “
indeterminate” X0 = 1, so 00 = 1
00 = 0, so 00 = 0
When in doubt
00 = “
indeterminate”
The example use “y and a” as
variables not x. It can be
changed to another alphabet
The question can be
solved using the formula
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GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
EXERCISE 1 “ Cogito Ergo Sum--Descrates”
1. Simlplify:
(a). 4
33
4.2
(d). 3a3 x 2ab2
(b). 62
33
3.)7(
9.)7(
(e). (2x3)-3
(c). r3 x r (d). s6 / s2
2. Simplify the expressions.
(a). ( ) ( ) ( )
( ) ( ) (b).
3. If x = 38, express x½ in the form 3n where n is an integer.
4. Given a = 29 x 5-6 Express a1/3 and a-1 in the form 2m x 5n where m and
n are integers.
5. The number 450 can be written as 2a x 3b x 5c.
Calculate the values of a, b, and c
B. EXPONENTIAL EQUATIONS
An exponential equation is an equation that contains a
variable with an exponent, or a variable in an exponent. For
example 5x = 5
3, 10
1–x = 10
4.
To solve exponential equations, you need to have equations
with comparable exponential expressions on either side of the
“equals” sign, so you can compare the power.
Let * + (
) If then m = n.
Did You Know Real Number
R
{
}
Q Rational Number
{
}
Z Integer
* +
Whole Numbers
Natural
Numbers
* +
* +
{ √ }
Irational
Number
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GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
Rule
If then {
Take a look at the example.
Solve
1. 32x–1 = 27 2. = 8.
32x–1 = 27 24x2+4x
= 23
32x–1 = 33 4x2 + 4x = 3
2x – 1 = 3 4x2 + 4x – 3 = 0
2x = 4 (2x – 1)(2x + 3) = 0
x = 2 x = 1/2 ,
–3/2
EXERCISE 2 “ Mathematics is universal language”
Find the solution of the exponential equations below
1. (a). If then x = … (b).If ( ) then x = ……
2.
3.
4. ( )
5.
6.
7.
8. Solve the equation
9. The formulae for the volume and the surface area of a sphere are V =
and respectively, where r is the sphere’s radius. Find
expressions for
(a). S in terms of V (b). V in terms of S
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GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
10. The moving kinetic energy, K joules, possessed by an object of mass
m kg moving with speed v ms-1 is given by the formula K =
Find
the kinetic energy possessed by a bullet of mass 1.5 x 10-3 kg moving
with speed 12 x 103 ms-1.
1.2 SURDS
Well , having been exposed to indices in the previous lessons will help you
understand the use of surds. In this section, we use the word “surd”
means not only a number with radical sign. Numbers whose square roots
cannot be determined in terms of rational numbers eg. √ etc are called
surds. Such numbers occur frequently in Trigonometry when calculating
the ratio of angles eg. Cos 30= √
, tan 60 = √ and in coordinate geometry
in the calculation of distances. Therefore You will find it useful to have a
sound knowledge of surds. Now lets begin our journey in SURDS.
A. UNDERSTANDING SURDS
What is the exactly result of 43 and
You get the result and 24
You can’t expressed exactly 3 (it is a surd) but 4 can be simplified (it
equals 2). 4 isn’t a surd.
Can you define the surd
Sometimes it is useful to work in surds, rather than using an approximate
decimal value. Surds can be manipulated just like algebraic expressions
and sometimes it may be possible to eliminate the surd (called
rationalizing the expression), which may have not been possible if you
tried to work with approximate value. When asked to give the
approximate decimal answer will not do and you will have to manipulate
surds in order to give final answer in simplified surd form.
Need more example ?
Have a look at these:
Number Simplifed As a Decimal Surd or
not?
√ √ 1.4142135(etc) Surd
√ √ 1.7320508(etc) Surd
√ 2 2 Not a surd
A Surd is an expression containing a root with an irrational solution that can
not be expressed exactly
History
How did we get the
word "Surd" ?
Well
around 820 AD al-
Khwarizmi (the
Persian guy who we
get the name
"Algorithm" from)
called irrational
numbers "'inaudible"
... this was later
translated to the Latin
surdus ("deaf" or
"mute")
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GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
√
0.5 Not a surd
√
√
2.2239800(etc) Surd
√
2 2 Not a surd
√
√
1.2457309(etc) Surd
As you can see, the surds have a decimal which goes on forever without
repeating, and that makes them irrational number. It is called irrational
because it cannot be written as a ratio (or fraction), not because it is
crazy . But if it is a number, it can be written as a simple fraction then
it is called a rational number.
The conclusion:
Laws of Surds
If it is a root and irrational, it is a surd.
But not all roots are surds.
History
Irrational Numbers
Apparently Hippasus (one of Pythagoras' students) discovered irrational numbers when trying to represent the square root of 2 as a fraction (using geometry, it is thought). Instead he proved you couldn't write the square root of 2 as a fraction and it was irrational.
However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove Hippasus' "irrational numbers" and so Hippasus was thrown overboard and drowned! Oh…. Dear!
However
Pythagoras
could not accept the
existence of irrational
numbers, because he
believed that all numbers
had perfect values. But he
could not disprove
If a, b, (a ≠ 0 and b ≠ 0 are positive) and (m and n are real numbers)
then
Multiplication of Surds :
1. abbxabxabxa 2
12
1
2
1
Division of Surds
b
a
b
a
b
a
b
aba
2
1
2
1
2
1
:.2
Addition of Surds:
anmanam .3 4. √ √ = √ √ (where a = m + n and b = m. n)
√ √ = √ - √ ( )
Substraction of Surds:
anmanam .5
Applying
mmm bxabxa
Applying m
m
m
b
a
b
a
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GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
To make you easy using surds, follow these steps:
1. Check which numbers are of the same base and which are of the
same power.
2. Use appropriate formulate.
Note : a means the positive square root of a, while - a means
the negative square root of a.
Have a look at the following examples:
1. Evaluate the following surds without the use of a calculator
(a). 182 x (b) 6
5
3
40
Solution:
(a). 636)8)(2(182 x (ans)
(b). 4165
6
3
40
6
53
40
6
5
3
40
(ans)
2. Evaluate the following surds, without the use of a calculator.
(a) 33
23
2
1 (b) 5354
Solution:
(a) 36
73)
3
2
2
1(3
3
23
2
1 (ans)
(b). 55345354 (ans)
The question can be solved using the formula
b
aba
The question can be solved using the formula
abbxa
The question can be solved using the formula
anmanam
The question can be solved using the formula
anmanam
√
√
√
√
√
√
√
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GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
EXERCISE 3
“ Plan for tomorrow but live for today”
1. Write as simply as possible
(a). 40290
(b). 402903
(c). 40390 x
2. Simplify the expression
(a). √ √ √
(b). √ √ √
(c). √
√
√
3. ABCD is a rectangle in which AB = √ cnd BC = √ . Giving each
answer in simplified form, find
(a). The area of the rectangle (b). The length of the diagonal AC
4. Find the length of the third side in each of the following right-
angled triangles, giving each answer in simplified surd form.
(a). √ (b). √ cm
cm 7 cm
5. Simplify the following questions
(a). √ √
(b). √ √ √ √
(c). √ √ √ √
√
6. Let a, b, m and n be four real numbers, satisfying a = m + n and
b = m. n. Then, proof that
(a). √ √ √ √
(b). √ √ √ √ , (m > n)
Did You Know
√ and
have different
meanings in the set
of all real numbers.
√ √ = 4
If then
or
√
Isaac Newton (1642-1727)
Isaac Newton is thought by many to have been one of the greatest intellects of all time. He went to Trinity College Cambridge in 1661 and by the age of 23 he had made three major discoveries: the nature of colours, the calculus and the law of gravitation. He used his version of the calculus to give the first satisfactory explanation of the motion of the Sun, the Moon and the stars. Because he was extremely sensitive to criticism, Newton was always very secretive, but he was always very secretive, but he was eventually persuaded to publish his discoveries in 1687.
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GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
7. (a). Find x, so that √ √ √ √
(b) Evaluate √ √ √ √
B. RATIONALISATION OF DENOMINATORS
Another technique to simplify expressions involving surds is to rationalise
the denominator. This means removing a surd from a fraction.
You can rationalize the denominator easily, just follow two kinds of
method below. Lets check it out !
Well, you have already known the steps. Practice will makes you
become perfect so lets look the examples before doing the
exercise
1. Rationalise the denominator of the following surds:
(a). 11
2 (b)
25
3
(c).
625
1
Solution:
(a). 1111
9
11
11
11
2
11
2
(b).
45
)25(3
25
253
25
25
25
3
25
3
22
1. Denominator has a single-term surd
If the denominator has a surd of the form a
Step : Rationalise it by multiplying the numerator and the denominator by
a to get
2. Denominator has sum or difference of surds
If the denominator is of the form anam
Step : Rationalise it bymultiplying the numerator and the denominator by its
conjugate surd anam to get a rational number m2a – n2b.
Puzzle
Math Aptitude Test
1. Take two apples from three apples and what do you have?
2. Some months have 30 days, some have 31; how many have 28 days?
Solution:
1. You have two apple
2. All of the months
have 28 days.
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GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
(c). 2425
625
625625
625
625
625.
625
1
EXERCISE 4 “ What oxygen is to lungs, such is hope to meaning life”
1. Rationalize the denominator in each of the following expression and
simplify them
a.
√ b.
√
√
2. Rationalize the denominator and simplify these fractions.
a.
√ c.
√
√ √
b. 643
1
d.
37
3
3. Evaluate √ √ √ √
√
4. a. Explain why
√
√
√
√ and hence show that
√
√
b.Show that
√ √
√ √
5. Find x if √√
√
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GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
1.3 LOGARITHM
Hmmm, hello… ..How are you doing?
In the previous lesson, you have learned about indices, aren’t you?. How is
it going? All of them seem easy for you? Can you solve this . Is it
a kind of indices? The solution remain a mystery, don’t it?
Well. What we are going to learn now is answering that mystery.
Logarithm answer your curiosity related to indices. I believe that all of
you ever think this kind of question.
In its simplest form, a logarithm answers a simple question above
Example
The question is an easy question and every one have the same idea to
answer that. But, how to write it?
We would write "the number of 2s you need to multiply to get 8 is 3" as
log2(8) = 2log(8) = 3
So, it means these two things are same:
Base
The number we are multiplying is called the "base", so we would say:
"the logarithm of 8 with base 2 is 3"
or "log base 2 of 8 is 3"
or "the base-2 log of 8 is 3"
So a logarithm also answers the question
What exponent do we need
(For one number to become another number)?
For the next,
we will write alogb , where
“a” is the base
How many of one number do we multiply to get another number?
How many 2 need to be multiplied to get 8?
Answer: 2 x 2 x 2 = 8, so we needed to multiply 3 of the 2 to get 8
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GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
Have a look at this example:
1.What is 10log (100) ……?
Ok. To answer the this question, you can back to our first question at
the first session in logarithm part. “What exponent do we need
(For 10 to become 100)?”
Answer:
102 = 100, so an exponent of 2 is needed to make 10 into 100.
It means that, 10log (100) = 2
2. What is 3log (81)…?
Answer:
34 = 81, so an exponent of 4 is needed to make into 8.
3log 81 = 4
3. If 7,05log x show that 7 355x
Answer:
)(55
55
5
5
57,05log
7 3
7
3
7
107
10
10
7
10
7
7,0
qedx
xx
xx
x
xx
So, After looking the apperception above
a logarithm is the exponent to which the base must be raised to produce
a given number
Based on the previous explanation, we can define a number as
DEFINITION
History
― …il cessa de calculer et de vivre — … he ceased to calculate and to live ‖.
Leonhard Paul
Euler April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. Euler is considered to be the preeminent mathematician of the 18th century and one of the greatest of all time. He is also one of the most prolific; his collected works fill 60–80 quarto volumes.[3] A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all. A Logarithm is the exponent to which the base must be raised to produce
a given number
agifonlyandifxa xg log
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GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
EXERCISE 5 “When life give you 100 reasons to cry, then show life that you have 1000 reasons to smile”.
1. Write in exponential form:
a). 664log2 d). 6125log5
b). 327
1log3 e). 23log3
c). 38
1log2
1
f). 201,0log10
2. Find the value of the following.
a). 32log2 f). 121
1log11
b). 49log7 g). 1000log
c). 1log3 h). 25log5
d). 243
1log3 i). 3log4
4
e). 11log13 j). 4log3
3
1
3. a). If 4,02log x show that 28
1x .
b). if 2
122log2
1
x , show that 24
3x
Common Logarithms : Base 10
Sometimes you will see a logarithm written without a base, like this :
Log(100). This usually means that the base is really 10
Common Logarithms : Base “e”
It is called a "common logarithm". Engineers love to use
it.
On a calculator it is the "log" button.
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GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
Another base that is often used is e (eulers number) which is
approximately 2, 71828.
This is called a "natural logarithm". Mathematicians use
this one a lot.
On a calculator it is the "ln" button.
It is how many times you need to use "e" in a multiplication, to get the
desired number.
Example: 389,771828,22389,7log389,7ln 2 becausee
You have learned various rules for manipulating and simplifying
expressions with exponents, such as the rule that says that x3 · x5 equals
x8 because you can add the exponents. There are similar rules for
logarithms.
Note: g is the base of logarithm which is satisfied 0 < g < 1 or g >1
(g > 0 and
g 1).
Before continuing the next journey about logarithm, it will be better for
you to see where do the rules of logarithm above come from.
Log Rules:
ag
aaiii
an
maii
bbai
gaii
g
aai
axna
bab
a
babxa
a
gng
gmg
gag
a
g
p
pg
gng
ggg
ggg
g
n
n
log.6
loglog).
loglog).
loglog.log)..5
log
1log).
log
loglog)..4
loglog.3
logloglog.2
loglog)log(.1
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GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
Can you imagine , if you do not know where do you come from, who is
your parents, where is your home town. You will feel that there is
something less in your life. It same like knowing the mathematics formula
without knowing how could be like that.
So, Lets notice following evidence
Rule 1 :
Prove: Suppose xag log maka xga ………..(1)
ybg log maka ygb ………..(2)
Multiply equation(1) to (2), we get:
yx gxgbxa
yxgbxa
yxgg gbxa log)(log
yxbxag )(log …………def logarithm
babxa ggg loglog)(log ………….(qed)
Rule 2 :
Prove : Divide equation 1 by equation 2, we get:
y
x
g
g
b
a
yxgb
a
yxb
ag
log
bab
a ggg loglog)(log ………….(qed)
Rule 3 :
Prove : axaxxaxaxaa gng ...loglog
n factor for each a
= aaaa gggg loglog...loglog
n times
babxa ggg loglog)log(
bab
a ggg loglog)log(
axna gng loglog
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GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
= n x ag log …………….(qed)
Ok. You have read some of the evidence of logarithm’s rules. You should
know all of them not only part of them,, so we give what you need to
improve your knowledge. Lets go!
Rule 4 :
Prove :
g
ax
gxa
ga
gathenxaSuppose
p
p
pp
xpp
xg
log
log
loglog
loglog
,log
).....(..........log
loglog qed
g
aa
p
pg
Substitute p = a at the right side, we get:
ga
g
aa
a
g
a
ag
log
1log
log
loglog
Rule 5:
Prove : i) a
bx
g
abxa ag
log
log
log
logloglog
ga
becomeequationtheapIf
b
aa
a
g
p
pg
log
1log
:,
log
loglog
The Rule 5 are the expand of the
previous rules
aaiii
an
maii
bbxai
gng
gmg
gag
n
n
loglog).
loglog).
logloglog).
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GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
g
b
log
log
bg log …………… (qed)
ii). n
mmg
g
aa
n
log
loglog
an
m
g
a
n
m
g log
log
log
……………. (qed)
iii) We leave the prove for you.
Do not think that you can’t ! Just do it. We believe you.
Rule 6 :
Prove : Suppose agthenxa xg ,log . I want you remember the
definition of logarithm.
So, thenxag ,log
ag
gg
a
xa
g
g
log
log
……………. (qed)
You should expect to need to know these rules, because there is a certain
type of question that the teacher can put on the test to make sure you
know how to use the rules; you won't be able to "cheat" with your
calculator. Here's what they look like:
Let blog(2) = 0.3869, blog(3) = 0.6131, and blog(5) = 0.8982. Using
these values, evaluate blog(10).
Since 10 = 2 · 5, then:
blog(10) = blog(2 · 5) = blog(2) + blog(5)
Since I have the values for blog(2) and blog(5), I can evaluate:
blog(2) + blog(5) = 0.3869 + 0.8982 = 1.2851
Then blog(10) = 1.2851.
Let blog(2) = 0.3869,blog(3) = 0.6131, and blog(5) = 0.8982. Using
these values, evaluate blog(7.5).
Rule 6 is the expand of logarithm
definition
ag ag
log
21
GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
This one is a bit more complicated. But if you take a little time,
you will notice that 7.5 = 15 ÷ 2, so:
blog(7.5) = blog(15 ÷ 2) = blog(15) – blog(2)
And 15 = 5 · 3, so: Copyright © Elizabeth Stapel 2000-2007 All
Rights Reserved
blog(15) – blog(2)
= [blog(5) + blog(3)] – blog(2)
= blog(5) + blog(3) – blog(2)
And now I can evaluate:
blog(5) + blog(3) – blog(2)
= 0.8982 + 0.6131 – 0.3869
= 1.1244
Then blog(7.5) = 1.1244.
EXERCISE 6 “Never give up, never give in, and don’t let your weakness
win”
1. Find the value of :
(a). 6log18log 33 (f). 6log23log24log 777
(b). 4
11log5log 22 (g). 9log1 2
1
(c). 4log3320log 55 (h). 21log
81log
(d).
2
12log
2log5log 22 (i). 13log254log
(e) 5,2log4,0 (j). 4,2log2log6log 555
2. Simplify
(a) aa gg loglog 4 (c). )1log()1log( 2 aa gg
(b). 2
1log
1loglog
xxx (d) 23 logloglog xxx xxx
3. If x = 0,6666… and b = 0,4444….
22
GRADE X
SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think
Find the value of 2log
2
yx.
4. If x
xxf
log.21
log)(
3
3
, find f(x) + f ...
3
x
5. If log 2 = a and l og 3 = b, find:
(a). log 18 (b) 5log 6
6. Evaluate the value of :
8log.54log.72log.3
512log.8
7. Find the value of a, if it is known:
6
5
4
1log16log8log 3 aaa
8. Find the value of x which is satisfied equation:
(a). 6log)1(log)2(log xx
(b). 1)1(log)1(log 33 xx
9. Given that : log 2 = 0,301 , log 3 = 0,477 dan log 7 = 0,845. Evaluate :
(a). log 5 (c) log 6
(b) 3 5,10log (d) log 7
6
10. Using properties of logs, show that
1
4
1ln4ln