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Mathematics Stage 5
PAS5.3.8
Part 3
Functions and logarithms
Logarithms
Number: 43668
Title: Functions and Logarithms
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Extracts from Mathematics Syllabus Years 7-10 © Board of Studies, NSW 2002
Unit overview pp iii,
iv, Parts 1, 2, 3
pp 3, 4
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Contents – Part 3
Introduction – Part 3 ..........................................................3
Preliminary quiz.................................................................5
Laws of logarithms ..........................................................11
Systems of logarithms.............................................................17
Further laws of logarithms ...............................................21
Using logarithm laws .......................................................27
Solving index equations ..........................................................30
Exponentials and logarithms ...........................................33
Exponential graphs .................................................................33
Logarithmic graphs..................................................................35
Relationship between curves..................................................37
Miscellaneous exercises .................................................41
Suggested answers – Part 3 ...........................................47
Exercises – Part 3 ...........................................................55
Part 3
Logarithms
1
2
PAS5.3.8 Functions and Logarithms
Introduction – Part 3
In 1935 the American seismologist Charles F. Richter (1900-1985)
recognised that the seismic waves radiated by all earthquakes can provide
good estimates of their magnitudes. He collected the recordings of
seismic waves from a large number of earthquakes, and developed a
calibrated system of measuring them for magnitude.
Richter showed that, the larger the intrinsic energy of the earthquake, the
larger the amplitude of ground motion at a given distance. He developed
a formula to calculate this magnitude:
M = log10 A + 3log10 (8∆t) − 2.92
where A is the maximum amplitude of the earthquake wave (in
millimetres), and ∆t is the time difference (in seconds) between the
arrival of the primary and secondary waves. ∆t (‘delta-t’) is used to
compensate for the distance from the source of the earthquake where the
measurement is made.
For example, if the maximum wave amplitude, A, is 23 mm and there is a
25 s gap ( ∆t ) between the arrival of the two waves, then
M = log10 23 + 3log10 (8 × 25) − 2.92
= 5.34
An earthquake whose magnitude is greater than 4.5 on this scale can
cause damage to buildings and other structures. Severe earthquakes have
magnitudes greater than 7. The famous San Francisco earthquake of
1906 recorded 7.8 on the Richter scale.
An undersea earthquake caused the Boxing Day tsunami of 2004. It
killed more than 283 100 people, making it one of the deadliest disasters
in modern history. The magnitude of the earthquake, the second largest
earthquake ever recorded, was estimated at 9.15 on the Richter scale.
This is just one example of the use of logarithms.
Part 3
Logarithms
3
Students learn about:
•
deducing a number of laws and relationships of logarithms
•
applying the laws of logarithms to evaluate simple expressions
•
simplifying expressions using the laws of logarithms
•
drawing the graph of y = log x and relating it to its inverse function
•
solving simple equations that contain exponents or logarithms.
Students learn to:
•
compare and contrast a set of exponential and logarithmic graphs
drawn on the same axes to determine similarities and differences.
Source:
4
Extracts from Mathematics Years 7–10 syllabus
© Board of Studies NSW, 2002.
PAS5.3.8 Functions and Logarithms
Preliminary quiz
Before you start this part, use this preliminary quiz to revise some skills
you will need.
Activity – Preliminary quiz
Try these.
1
Solve the following.
a
3m − 2 = 27
___________________________________________________
___________________________________________________
___________________________________________________
b
251− y = 1251+ y
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
2
Write the following in index form.
a
log 7 49 = 2
___________________________________________________
b
2
3
= log 8 4
___________________________________________________
___________________________________________________
Part 3
Logarithms
5
3
Write the following in logarithmic form.
a
161.5 = 64
___________________________________________________
81 = 9
b
___________________________________________________
___________________________________________________
c
4
625 = 5
___________________________________________________
d
3−2 =
1
9
___________________________________________________
___________________________________________________
4
Evaluate.
a
log 2 8
___________________________________________________
___________________________________________________
___________________________________________________
b
log 8 4
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
c
log 9 9
___________________________________________________
___________________________________________________
___________________________________________________
6
PAS5.3.8 Functions and Logarithms
d
log 8 1
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
5
Find the value of the pronumeral.
a
log x 2 = 1
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b
log m 32 = 5
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
c
log n
1
= −3
27
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Part 3
Logarithms
7
d
log v
2
1
=−
3
4
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
e
log w 9 =
2
3
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
f
log 5 d = 4
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
g
log10 j = 2.5
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
8
PAS5.3.8 Functions and Logarithms
6
Evaluate.
a
log 36 3 6
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b
log 49 49 7
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
Part 3
Logarithms
9
10
PAS5.3.8 Functions and Logarithms
Laws of logarithms
You have already learned the laws for indices. When you remember that
logarithms are, in fact, indices:
the logarithm of a number to a given base is the index to which you
raise that base to get the number
then you can realise that the laws for logarithms are the laws for indices
written in logarithmic form.
Index law 1
Logarithm law 1
a0 = 1
log a 1 = 0
Any base to the power of 0 is equal to
1. (a ≠ 0)
The logarithm of 1, to any base, is
equal to zero.
Examples
Examples
20 = 1
n0 = 1
log 2 1 = 0
log 5 1 = 0
log n 1 = 0
Index law 2
Logarithm law 2
a = a1
log a a = 1
Any number raised to the power of 1
has the same value.
The logarithm of a number to the
same base is always equal to 1.
Examples
Examples
50 = 1
3 = 31
10 = 10
1
x = x1
log 3 3 = 1
log10 10 = 1
log x x = 1
You should now be able to use these two laws to write down the answers
to the following.
Part 3
Logarithms
11
Activity – Laws of logarithms
Try these.
1
Write down the value of x in each of the following.
a
log 4 x = 0 __________________________________________
b
log 3 1 = x ___________________________________________
c
log 4 x = 1 __________________________________________
d
log x 8 = 1 __________________________________________
Check your response by going to the suggested answers section.
Could you immediately write down
the value of x in each of these without
needing to work them out?
Index law 3
Logarithm law 3
am × an = am + n
log a (M × N ) = log a M + log a N
To multiply powers of the same base,
add the indices.
The log of a product of two numbers is
the sum of the logs of those numbers
to the same base.
Now study carefully how Index law 3 leads you to the third law for
logarithms.
Let M = a m andN = a n .
By the definition of logarithms, m = log a M andn = log a N .
M × N = am × an
∴MN = a m + n
12
PAS5.3.8 Functions and Logarithms
This means,
log a (MN ) = m + n
∴log a (MN ) = log a M + log a N
Examples
log 3 35 = log 3 (5 × 7) = log 3 5 + log 3 7
log a 40 = log a (4 × 10) = log a 4 + log a 10
log a 40 = log a (5 × 8) = log a 5 + log a 8
Follow through the steps in this example. Do your own working in the
margin if you wish.
If log a 2 = 0.301,log a 3 = 0.477andlog a 5 = 0.699 ,
calculate the value of
a
log a 6
b
log a 15
Solution
a log a 6 = log a (2 × 3)
= log a 2 + log a 3
= 0.301 + 0.477
= 0.778
b
log a 15 = log a (3 × 5)
= log a 3 + log a 5
= 0.477 + 0.699
= 1.176
As you are only given the logs of 2, 3, and 5, you need to write 6 and 15
as products of these numbers to work them out.
Part 3
Logarithms
13
Activity – Laws of logarithms
Try these.
2
Given log a 2 = 0.301,log a 3 = 0.477andlog a 5 = 0.699 , calculate
the value of
a
log a 4
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b
log a 20
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
3
Show that log a 9 is double log a 3 . Hence calculate its value.
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
In these examples it does not matter what the base is, as you were
provided with the value of the logarithm.
14
PAS5.3.8 Functions and Logarithms
From these examples you should
realise that you can extend
logaMN = logaM + logaN
to
logaMNP = logaM + logaN + logaP
and so on.
And remember, these rules only
apply if the bases remain the same.
Law 3: log a (M × N ) = log a M + log a N can also be written as
log a M + log a N = log a (M × N ) .
This states that
The sum of the logarithms of two numbers, to the same base, is the
logarithm of the product of the numbers to the same base.
This rule can be extended to more than just two numbers.
Examples
log10 3 + log10 8 = log10 (3 × 8) = log10 24
log 2 6 + log 2 7 = log 2 (6 × 7) = log 2 42
log a 2 + log a 5 + log a 7 = log a (2 × 5 × 7) = log a 70
Sometimes you arrive at a result where you might be able to evaluate the
logarithm using either of laws 1 or 2.
Follow through the steps in this example. Do your own working in the
margin if you wish.
Evaluate by first writing as a single logarithm.
a
log 6 2 + log 6 3
b
log15
c
log10 16 + log10 5 + log10 125
1
+ log15 30
2
Solution
Part 3
Logarithms
15
a
log 6 2 + log 6 3 = log 6 (2 × 3)
= log 6 6
= 1[Logarithm law 2]
b
log15
c
log10 16 + log10 5 + log10 125
1
⎛1
⎞
+ log15 30 = log15 ⎜ × 30 ⎟
⎝2
⎠
2
= log15 15
= 1[Logarithm law 2]
= log10 (16 × 5 × 125)
= log10 10000
Now this single logarithm can be expanded
log10 10000 = log10 (10 × 10 × 10 × 10)
= log10 10 + log10 10 + log10 10 + log10 10
= 1+1+1+1
=4
Make sure you are familiar with the logarithm
laws so far before attempting the next activity.
Activity – Laws of logarithms
Try these.
4
5
Write as a single logarithm.
a
log 4 7 + log 4 3 =
b
log b 10 + log b 9 = ____________________________________
c
log 5 3 + log 5 2 + log 5 11 =
____________________________________
_____________________________
Evaluate by first writing as a single logarithm.
a
log18 6 + log18 3 =
___________________________________________________
___________________________________________________
16
PAS5.3.8 Functions and Logarithms
b
log 24 2 + log 24 3 + log 24 4 =
___________________________________________________
___________________________________________________
c
log10 2 + log10 50 =
___________________________________________________
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
Systems of logarithms
Any positive number, other than 1, can be used as the base for a system
of logarithms (to that base).
There are two systems in wide current use.
•
Natural logarithms
Sometimes called Naperian logarithms, after John Napier (15501617). These are important in senior and higher mathematics and
have as the base the irrational number e = 2.71828... . You will not
be using these logarithms in this course.
•
Common logarithms
These are also called Briggsian logarithms and use the base 10.
They are convenient for computation in our decimal number system.
Part 3
Logarithms
17
Both of these two systems may be found
on your calculator.
The log key gives logarithms to the
base 10, and the ln key is used for
logarithms to the base e. (ln stands for
logarithme naturale, natural logarithm).
So if you wanted to find log10 27 , you
would press log 27.
The answer your calculator should give
is 1.431363764
When you use the log key your answer
won’t always be a nice whole number.
But for now you will not need your calculator.
By convention, if you are working with common logarithms, that is in
base 10, you can leave out the base. For example, log10 n can be simply
written as log n. (log 9 means log10 9 and log 4 means log10 4 , and so
on.)
Follow through the steps in this example. Do your own working in the
margin if you wish.
Given that log 9 = 0.9542, find the value of log 90.
Solution
log 90 = log10 90
= log10 (9 × 10)
= log10 9 + log10 10
= 0.9542 + 1[Notelog10 10 = 1]
= 1.9542
You can check this answer with your calculator.
18
PAS5.3.8 Functions and Logarithms
Activity – Laws of logarithms
Try these.
6
Given log 15 = 1.1761, find log 150.
_______________________________________________________
_______________________________________________________
7
Given log 2 = 0.3010 and log 3 = 0.4771, find log 60.
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
You have been practising using some laws of logarithms. Now check
that you can solve these kinds of problems by yourself.
Go to the exercises section and complete Exercise 3.1 – Laws of
logarithms.
Part 3
Logarithms
19
20
PAS5.3.8 Functions and Logarithms
Further laws of logarithms
You are familiar with this index law: a m ÷ a n = a m − n .
Using M = a m andN = a n (that is, m = log a M;n = log a N ) then
M
= M ÷ N = am−n .
N
⎛M⎞
And from the definition of logarithms, log a ⎜ ⎟ = m − n .
⎝N⎠
⎛M⎞
That is, log a ⎜ ⎟ = log a M − log a N
⎝N⎠
Index law 4
Logarithm law 4
am ÷ an = am − n
⎛M⎞
log a ⎜ ⎟ = log a M − log a N
⎝N⎠
To divide powers of the same base,
subtract the indices.
The log of the quotient of two numbers
is the difference of the logs of those
numbers to the same base.
Examples
15
= log a 15 − log a 3
3
log b (20 ÷ 7) = log b 20 − log b 7
log a
log p 0.75 = log p
3
= log p 3 − log p 4
4
Conversely, you can also say
The difference in the logarithms of two numbers to the same base
is the logarithm of the quotient of the two numbers to the same
base.
Example
log a 7 − log a 2 = log a
Part 3
Logarithms
7
= log a 3.5
2
21
Follow through the steps in this example. Do your own working in the
margin if you wish.
a
Given log10 2 = 0.301andlog10 5 = 0.699 , calculate
log10 0.5 .
b
Simplify log 3 24 − log 3 8
Solution
a
b
log10 0.5
5
= log10
10
= log10 5 − log10 10
= 0.699 − 1
= −0.301
log10 0.5
1
= log10
2
= log10 1 − log10 2
= 0 − 0.301
= −0.301
OR
⎛ 24 ⎞
log 3 24 − log 3 8 = log 3 ⎜ ⎟
⎝ 8⎠
= log 3 3
=1
You should realise that just as
loga MN = loga M + loga N
then
loga M
= loga M – loga N
loga N
22
PAS5.3.8 Functions and Logarithms
Activity – Further laws of logarithms
Try these.
1
You are given that log 3 = 0.477, log 4 = 0.602 and log 5 = 0.699.
[As no base is given, assume it to be 10.] Calculate
a
log
3
= ____________________________________________
4
___________________________________________________
___________________________________________________
___________________________________________________
b
log 0.8 = ____________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
2
How can you use log
3
to help you find the value of log 75? Show
4
your working.
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
There is just one more logarithm law you need to know.
Part 3
Logarithms
23
Consider
log a x n = log a (x × x × x × x × ....... × x)
= log a x + log a x + ....... + log a x
= n log a x
[x is a factor n times]
[ log a x is a term added n times]
Index law 5
Logarithm law 5
a n = a×
a ×
a ×
... ×a
log a x n = n log a x
there are n factors altogether
To raise a number to a power, multiply
the number by itself the index number
of times.
The logarithm of the nth power of a
number is n times the logarithm of the
number to the same base.
For instance, log 2 5 8 = 8 log 2 5 .
Follow through the steps in this example. Do your own working in the
margin if you wish.
Given log a 3 = 0.5 , find the value of
a
log a 81
b
log a
1
9
Solution
log a 81
a
b
= log a 34
= 4 log a 3
= 4 × 0.5
=2
1
log a
9
= log a 3−2
= −2 log a 3
= −2 × 0.5
= −1
Because you were given the value of loga 3,
you needed to write loga 81 and loga 1
2
in terms of loga 3 to evaluate them.
24
PAS5.3.8 Functions and Logarithms
Activity – Further laws of logarithms
Try these.
3
Given that log a 2 = 0.3010 find the value of
a
log a 8 = ________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b
log a 64 = ___________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
You have been practising applying further rules of logarithms. Now
check that you can solve these kinds of problems by yourself.
Go to the exercises section and complete Exercise 3.2 – Further laws of
logarithms.
Part 3
Logarithms
25
26
PAS5.3.8 Functions and Logarithms
Using logarithm laws
Here is a summary of the laws of logarithms and relationships you have
learned so far.
•
log a 1 = 0
•
log a a = 1
•
log a (M × N ) = log a M + log a N
•
⎛M⎞
log a ⎜ ⎟ = log a M − log a N
⎝N⎠
•
log a x n = n log a x
Make sure you understand and can use
these logarithm relationships before
proceeding further.
The laws of logarithms help you simplify expressions such as the
following.
Follow through the steps in this example. Do your own working in the
margin if you wish.
a
Simplify log x + log y – 2log z
b
Write 4log 2 – 2log 6 + 2log 3 – 3 log 4 as a single
logarithm.
Solution
log x + log y − 2 log z
a
= log x + log y − log z 2 [2 log z = log z 2 ]
x×y
z2
xy
= log 2
z
= log
Part 3
Logarithms
27
b
4 log 2 − 2 log 6 + 2 log 3 − 3log 4
= log 2 4 − log 6 2 + log 32 − log 4 3
2 4 × 32
62 × 4 3
16 × 9
= log
36 × 64
1
= log
16
= log 2 −4
= −4 log 2
= log
In these two examples no base was given. You could therefore assume it
to be 10. However, the base you use here doesn’t matter; the outcome is
still be the same.
Activity – Using logarithm laws
Try these.
1
Write as a single logarithm.
a
log xy + log yz − log xz
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b
4 log 27 + 2 log 2 − 2 log 3
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
28
PAS5.3.8 Functions and Logarithms
You can also proceed the other way and expand logarithmic expressions.
Follow through the steps in this example. Do your own working in the
margin if you wish.
Write log y
ax
in expanded form.
y2
Solution
log y
ax
= log y ax − log y y 2
2
y
= log y a + log y x − 2 log y y
= log y a + log y x − 2[log y y = 1]
Expand these step-by-step.
Activity – Using logarithm laws
Try these.
2
Write these in expanded form.
a
log b abc
___________________________________________________
___________________________________________________
___________________________________________________
b
log10
x
y
− log10
y
x
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
Part 3
Logarithms
29
Notice that in the second example it didn’t matter whether the base is 10
or any other base. The result is still the same.
Solving index equations
You are familiar with equations of the type: 8 = 2 2 x − 2 .
Its solution is based on the fact that you can equalise the bases easily.
8 = 22 x − 2
23 = 22 x − 2
3 = 2x − 2[same base, therefore equate indices]
2x = 5
5
x=
2
In the equation 2 x = 3 there is no simple method of equalising the bases.
Here is a way to solve such equations.
If 2 x = 3 then the common logarithms of both sides are equal. That is,
2x = 3
log10 2 x = log10 3
x log10 2 = log10 3
log10 3
x=
log10 2
= 1.5850[correct to 4 dp using a calculator]
For this next activity you will need to use your calculator.
30
PAS5.3.8 Functions and Logarithms
Activity – Using logarithm laws
Try these.
3
Solve 3n +1 = 12 . Give your answer correct to 4 decimal places.
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
4
Solve 5 y −1 = 3y . Give your answer correct to 4 significant figures.
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
You have been practising using logarithm laws to simplify expressions
and solving index equations with different bases. Now check that you
can solve these kinds of problems by yourself.
Go to the exercises section and complete Exercise 3.3 – Using logarithm
laws.
Part 3
Logarithms
31
32
PAS5.3.8 Functions and Logarithms
Exponentials and logarithms
Functions of the form y = a x are known as exponential functions.
Those of the form y = log a x are known as logarithmic functions.
Exponential graphs
Consider graphs of the type y = a x where a is a fixed, positive number
not equal to 1, and x is the index (or power or logarithm or exponent).
You may recall covering these curves earlier in your course. They are
called exponential graphs, or graphs of exponential functions.
Activity – Exponentials and logarithms
Try these.
1
Complete the table of values for y = 3x .
x
y
Part 3
Logarithms
–2
−2
3 =
1
2
3
–1
=
0
1
2
1
9
33
2
Draw the graph.
y
8
6
4
2
–2
–1
0
1
2 x
3
Where does the graph cut the y-axis? ________________________
4
Does the curve ever cut the x-axis? Give a reason.
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
This graph approaches the x-axis but never touches it. A line to which a
curve approaches gradually but never touches is called an asymptote.
The x-axis is an asymptote to this curve.
Here are some other features that you ought to be aware of about
exponential graphs.
•
When x = 0,y = a 0 = 1 . So the graph of y = a x always passes
through (0, 1)
•
When x = 1,y = a1 = a . So (1, a) is a point on the graph.
•
For a point (x0 ,y0 ) on the graph, y0 = a x0 . Now if you increase the
x-value by 1, the new y-value is given by
34
PAS5.3.8 Functions and Logarithms
y( x0 +1) = a x0 +1
= a1 × a x0 [by the law of indices]
= a × y0
So each time you add 1 to x, you multiply y by a.
•
Each time you subtract 1 from x, you divide y by a.
From these features you should easily recognise and be able to sketch the
graph of y = a x , for any values of a (≠1).
y
=
ax
y
a
(1, a)
1
0
1
x
Logarithmic graphs
y = log a x is a logarithmic function. What does a graph of this function
look like?
Follow through the steps in this example. Do your own working in the
margin if you wish.
Graph y = log 3 x
Solution
To draw up a table of values for this equation, it is easier to first
rewrite this relation as x = 3y .
In this case it is easier to give values for y and find the
corresponding values of x.
Part 3
Logarithms
35
y
–2
3−2 =
x
–1
1 1
=
32 9
3−1 =
1
3
0
1
2
30 = 1
31 = 3
32 = 9
[Notice how the table is set out; y-values first, so you can then calculate
the corresponding x-values.]
Plotting these points onto a grid gives:
y
3
2
1
–1
0
1
2
3
4
5
6
7
8
9
10
x
–1
–2
–3
From this graph you can estimate values of log 3 x . For example, when
x = 8, log 3 8 ≈ 1.9 .
Activity – Exponentials and logarithms
Try these.
5
Use the graph of y = log 3 x on the previous page to answer the
following.
a
36
Find
i
log 3 3
ii
log 3 1 __________________________________________
iii
log 3 5
_________________________________________
_________________________________________
PAS5.3.8 Functions and Logarithms
b
For what value of x is
i
log 3 x = 0.8 ? ____________________________________
ii
log 3 x = 1.6 ? ____________________________________
Check your response by going to the suggested answers section.
Here are some features that you ought to be aware of about logarithmic
graphs.
•
The curve lies totally to the right of the y-axis.
•
The y-axis is an asymptote to the curve y = log a x .
•
The curve cuts the x-axis at (1, 0), since log a 1 = 0 for all values of a.
•
The point (a, 1) lies on the graph y = log a x because log a a = 1 .
•
For x > 1, y = log a x is positive.
•
For 0 < x < 1, y = log a x is negative.
y
y=
log a
x
1
0
1
a
x
Relationship between curves
The sketch of the two curves y = a x andy = log a x on the same grid
shows that the figure is symmetrical.
Part 3
Logarithms
37
y
y = 3x
10
9
y=x
8
7
6
5
4
3
2
y = log3 x
1
–3
–2 –1 0
1
2
3
4
5
6
7
8
9
10 x
–1
–2
–3
In fact, each curve is a reflection of the other across the line y = x.
The exponential curve y = a x is a reflection of the logarithmic curve
y = log a x in the line y = x. Hence the exponential curve y = a x is the
inverse function of the logarithmic curve y = log a x .
Follow through the steps in this example. Do your own working in the
margin if you wish.
Find the area of the rectangle MNOP.
1
0
y
=
a
M
2x
8
y
P
N
x
Solution
38
PAS5.3.8 Functions and Logarithms
At the point M the value of y is 8. Substituting y = 8 into the
equation y = 2 x gives
y = 2 x
8 = 2 x
2 3 = 2 x
∴x = 3
The coordinates of M are (3, 8).
Hence ON = 3 units, MN = 8 units.
Therefore area of MNOP = 3 × 8 = 24units 2
Activity – Exponentials and logarithms
Try these.
6
In the sketch RS is parallel to OT. Find the coordinates of R, S and T.
y=
2x
y
S
y=
log
2
x
T
0
R
N
x
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Part 3
Logarithms
39
Check your response by going to the suggested answers section.
You have been practising comparing the relationship between
exponential and logarithmic curves. Now check that you can solve these
kinds of problems by yourself.
Go to the exercises section and complete Exercise 3.4 – Exponentials and
logarithms.
40
PAS5.3.8 Functions and Logarithms
Miscellaneous exercises
You are now in a position to deal with more difficult, and abstract,
examples of logarithms.
Text linking to example.
Follow through the steps in this example. Do your own working in the
margin if you wish.
Given log b 3 = mandlog b 2 = n , find the value of the
following in terms of m and/or n.
a
log b 16
b
log b 24
Solution
a
log b 16
= log b 2 4
= 4 log b 2
=4×n
= 4n
b
log b 24
= log b (3 × 8)
= log b 3 + log b 8
= m + log b 2 3
= m + 3log b 2
= m + 3n
Part 3
Logarithms
41
Activity – Miscellaneous exercises
Try these.
1
Given log b 3 = mandlog b 2 = n , find the value of the following in
terms of m and/or n.
a
log b 8b
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b
log b
12
b2
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
You should also be able to solve logarithmic equations.
In these examples you will be using the idea that if
log p = log q
then
p = q.
42
PAS5.3.8 Functions and Logarithms
Follow through the steps in this example. Do your own working in the
margin if you wish.
a
Write the value of x in terms of y if log 4x − log 3y = log 3 .
b
Solve the equation log 2x − log(x − 3) = log 3 .
Solution
log 4x − log 3y = log 3
a
4x
= log 3
3y
4x
=3
3y
4x = 9y
9
x= y
4
log 2x − log(x − 3) = log 3
2x
= log 3
log
x−3
2x
=3
x−3
2x = 3(x − 3)
2x = 3x − 9
−x = −9
x=9
log
b
In these examples it did not matter what the
base is. You can assume it to be 10, but the
relationship is still true with any other base.
Make sure you understand the logarithm relationships when working on
these.
Part 3
Logarithms
43
Activity – Miscellaneous exercises
Try these.
2
Express y in terms of x without the use of logs in
log 3x + log y = log 5 .
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
3
Solve the following: log x + log(2x + 1) = log(x + 4) .
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
Notice that in the last example x has to be
a positive number. This is because you can
only take the log of a positive number.
log x can only exist when x > 0.
Follow through the steps in this example. Do your own working in the
margin if you wish.
Simplify log 64 ÷ log16 .
44
PAS5.3.8 Functions and Logarithms
Solution
log 64 ÷ log16 =
log 64
log16
log 2 6
log 2 4
6 log 2
=
4 log 2
6
=
4
1
=1
2
=
It is important not to confuse log 64 ÷ log 16
with log 64 – log 16.
log
= log 64 – log 16
Activity – Miscellaneous exercises
Try these.
4
Simplify log a 4 ÷ log
( a) .
3
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
Part 3
Logarithms
45
You have been practising solving a variety of logarithmic expressions
and equations. Now check that you can solve these kinds of problems by
yourself.
Go to the exercises section and complete Exercise 3.5 – Miscellaneous
exercises.
46
PAS5.3.8 Functions and Logarithms
Suggested answers – Part 3
Check your responses to the preliminary quiz and activities against these
suggested answers. Your answers should be similar. If your answers are
very different or if you do not understand an answer, contact your teacher.
Activity – Preliminary quiz
1
a
3m − 2 = 27
b
(5 )
3m − 2 = 33
m−2= 3
m=5
2 1− y
( )
= 53
1+ y
5 2 − 2 y = 5 3+ 3 y
2 − 2y = 3 + 3y
5y = −1
1
y=−
5
2
3
2
a
7 2 = 49
b
8 =4
3
a
log16 64 = 1.5
b
log 81 9 =
c
log 625 5 =
d
log 3
a
Let x = log 2 8 .
b
Let x = log 8 4 .
4
1
4
Let x = log 9 9 .
9 x = 91
x =1
Logarithms
1
= −2
9
(2 3 )x = 2 2
2x = 23
x=3
c
1
2
8x = 4
2x = 8
Part 3
251− y = 1251+ y
2 3x = 2 2
3x = 2
2
x=
3
d
Let x = log 8 1 .
8x = 1
8 x = 80
x=0
47
5
a
log x 2 = 1
b
m 5 = 32
x1 = 2
x=2
c
log n
log m 32 = 5
1
= −3
27
1
n −3 =
27
−3
n = 3−3
n=3
d
m 5 = 2 5 (orm = 5 32 )
m=2
2
1
log v = −
3
4
2
−
1
v 3=
4
⎛ − 23 ⎞
⎜⎝ v ⎟⎠
−
3
2
⎛ 1⎞
=⎜ ⎟
⎝ 4⎠
−
3
2
3
−2 − 2
( )
v= 2
= 23
=8
e
log w 9 =
2
3
f
log 5 d = 4
d=
2
3
w =9
3
2
3
⎛ 23 ⎞
2
⎜⎝ w ⎟⎠ = ( 9 )
3
2 2
( )
w= 3
( 5)
⎛ 1⎞
= ⎜ 52 ⎟
⎝ ⎠
4
4
= 52
= 25
= 33
= 27
g
log10 j = 2.5
j = 10 2.5
= 10 2 × 10 0.5
= 100 × 10
= 100 10
48
PAS5.3.8 Functions and Logarithms
6
a
Let x = log 36 3 6 .
Let x = log 49 49 7 .
b
36 x = 3 6
(6 )
2 x
=6
49 x = 49 7
1
3
(7 )
2 x
=7 ×7
1
62 x = 6 3
1
2x =
3
1
x=
6
72 x = 7
2
2
1
2
1
2
1
2
1
x =1
4
2x = 2
Activity – Laws of logarithms
1
a
x=1
2
a
log a 4
= log a (2 × 2)
= log a 2 + log a 2
= 0.301 + 0.301
= 0.602
5
Part 3
x=0
c
b
x=4
d
x=8
log a 20
= log a (2 × 2 × 5)
= log a 2 + log a 2 + log a 5
= 0.301 + 0.301 + 0.699
= 1.301
log a 9
= log a (3 × 3)
= log a 3 + log a 3
= 2 log a 3(which is double log a 3)
= 2 × 0.477
= 0.954
3
4
b
a
log 4 7 + log 4 3
= log 4 (7 × 3)
= log 4 21
c
log 5 3 + log 5 2 + log 5 11
= log 5 (3 × 2 × 11)
= log 5 66
a
log18 6 + log18 3
= log18 (6 × 3)
= log18 18
=1
Logarithms
b
log b 10 + log b 9
= log b (10 × 9)
= log b 90
b
log 24 2 + log 24 3 + log 24 4
= log 24 (2 × 3 × 4)
= log 24 24
=1
49
c
log10 2 + log10 50
= log10 (2 × 50)
= log10 100
= log10 (10 × 10)
= log10 10 + log10 10
6
7
= 1+1
=2
log150 = log(15 ×10)
= log15 + log10
= 1.1761+1
= 2.1761
log 60 = log(2 × 3 × 10)
= log 2 + log 3 + log10
= 0.3010 + 0.4770 + 1
= 1.778
Activity – Further laws of logarithms
3
= log 3 − log 4
4
= 0.477 − 0.602
= −0.125
1
a
2
Start by writing log
log
b
4
5
= log 4 − log 5
= 0.602 − 0.699
= −0.097
log 0.8 = log
3
75
= –0.125.
= log
4
100
75
log
= −0.125
100
log 75 − log100 = −0.125
log 75 − log(10 × 10) = −0.125
log 75 − (log10 + log10) = −0.125
log 75 − (1 + 1) = −0.125(sincelog10 = log10 10 = 1)
log 75 = 2 − 0.125
= 1.875
3
50
a
log a 8 = log a 2 3
b
log a 64 = log a 2 6
= 3log a 2
= 3 × 0.3010
= 6 log a 2
= 6 × 0.3010
= 0.9030
= 1.8060
PAS5.3.8 Functions and Logarithms
Activity – Using logarithm laws
1
a
log xy + log yz − log xz
= (log x + log y) + (log y + log z) − (log x + log z)
= log x + log y + log y + log z − log x − log z
= 2 log y
b
log 27 + 2 log 2 − 2 log 3
= log 33 + 2 log 2 − 2 log 3
= 3log 3 + 2 log 2 − 2 log 3
= log 3 + 2 log 2
= log 3 + log 2 2
= log(3 × 2 2 )
= log(3 × 4)
= log12
2
a
log b abc = log b a + log b b + log b c
= log b a + 1 + log b c
b
3
log10
x
y
− log10 = (log10 x − log10 y) − (log10 y − log10 x)
y
x
= log10 x − log10 y − log10 y + log10 x
= 2 log10 x − 2 log10 y
3n +1 = 12
log10 3n +1 = log10 12
(n + 1)log10 3 = log10 12
n +1=
log10 12
log10 3
= 2.261859507[using a calculator]
n = 1.2619[correct to 4 dp]
[Notice that
Part 3
Logarithms
log10 12
≠ log10 4 . Don’t confuse this!]
log10 3
51
5 y −1 = 3y
4
log10 5 y −1 = log10 3y
(y − 1)log10 5 = y log10 3
y log10 5 − log10 5 = y log10 3
y log10 5 − y log10 3 = log10 5
y(log10 5 − log10 3) = log10 5
log10 5
log10 5 − log10 3
= 3.151[correct to 4 sig fig]
y=
Activity – Exponentials and logarithms
1
Table values are:
1 1
, ,1,3,9 .
9 3
y
2
8
6
4
2
–2
–1
0
2 x
1
3
(0, 1)
4
No, the curve never cuts the x-axis. As you move to the left the yvalue gets smaller and smaller but always remains positive.
52
5
a
b
i
i
1
2.4
6
R(1, 0), T(0, 1), S(1,2)
ii
ii
0
5.5
iii 1.5
PAS5.3.8 Functions and Logarithms
Activity – Miscellaneous exercises
1
a
b
log b 8b
= log b 8 + log b b
= log b 2 3 + log b b
12
b2
= log b 12 − log b b 2
log b
= log b (3 × 2 2 ) − 2 log b b
= 3log b 2 + 1
= log b 3 + log b 2 2 − 2 × 1
= 3n + 1
= log b 3 + 2 log b 2 − 2
= m + 2n − 2
2
log 3x + log y = log 5
3x
= log 5
log
y
3x
=5
y
3x = 5y
3
y= x
5
3
log x + log(2x + 1) = log(x + 4)
log x(2x + 1) = log(x + 4)
x(2x + 1) = x + 4
2x 2 + x = x + 4
2x 2 = 4
x2 = 2
x=± 2
x= 2
(since x must be positive)
log a 4 ÷ log
4
=
=
( a)
3
log a 4
⎛ 1⎞
log ⎜ a 2 ⎟
⎝ ⎠
3
log a 4
3
2
log a
4 log a
= 3
2 log a
3
2
2
=4×
3
2
=2
3
=4÷
Part 3
Logarithms
53
54
PAS5.3.8 Functions and Logarithms
Exercises – Part 3
Exercises 3.1 to 3.5
Name
___________________________
Teacher
___________________________
Exercise 3.1 – Laws of logarithms
1
Solve.
a
y = log 5 5 ___________________________________________
b
log 5 2x = 1
___________________________________________________
2
Given log a 2 = 0.301,log a 5 = 0.699, and log a 7 = 0.845, find the
value of
a
log a 14
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b
log a 35
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Part 3
Logarithms
55
c
log a 25
___________________________________________________
___________________________________________________
___________________________________________________
d
log a 49
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
3
a
Using the values in question 2, find the value of log a 10 .
___________________________________________________
___________________________________________________
b
What is the value of the base a? _________________________
Why? _____________________________________________
4
Write as a single logarithm.
a
log m 6 + log m 8
___________________________________________________
b
log b 9 + log b 4 + log b 5
___________________________________________________
___________________________________________________
5
Evaluate each of the following by first writing as a single logarithm.
a
log10 8 + log10 125
___________________________________________________
___________________________________________________
___________________________________________________
56
PAS5.3.8 Functions and Logarithms
b
log 6 4 + log 6 9
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
c
log12 3 + log12 36 + log12 16
___________________________________________________
___________________________________________________
___________________________________________________
d
log10 5 + log10 8 + log10 25
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
6
Given log 2 = 0.3010, log 3 = 0.477, and log 7 = 0.8451 evaluate
a
log 20
___________________________________________________
___________________________________________________
___________________________________________________
b
log 21
___________________________________________________
___________________________________________________
___________________________________________________
c
log 140
___________________________________________________
___________________________________________________
___________________________________________________
Part 3
Logarithms
57
d
log 42
___________________________________________________
___________________________________________________
___________________________________________________
7
a
Given that log 3 = 0.4771, evaluate log 5 + log 6.
___________________________________________________
___________________________________________________
___________________________________________________
b
Given that log 6 = 0.7782, evaluate log 5 + log 3 + log 4.
___________________________________________________
___________________________________________________
___________________________________________________
c
Given that log 4 = 0.6021, evaluate log 5 + log 8.
___________________________________________________
___________________________________________________
___________________________________________________
d
Given that log 9 = 0.9542, evaluate log 5 + log 6 +log 3.
___________________________________________________
___________________________________________________
___________________________________________________
e
Given that log 12 = 1.0792, evaluate log 15 + log 8.
___________________________________________________
___________________________________________________
___________________________________________________
58
PAS5.3.8 Functions and Logarithms
Exercise 3.2 – Further laws of logarithms
1
Given that log 2 = 0.301, log 3 = 0.477 and log 5 = 0.699 evaluate
a
log 0.2
___________________________________________________
___________________________________________________
___________________________________________________
b
log 2.5
___________________________________________________
___________________________________________________
___________________________________________________
c
log 1
2
3
___________________________________________________
___________________________________________________
___________________________________________________
d
log 7.5
___________________________________________________
___________________________________________________
___________________________________________________
2
Given that log a 2 = 0.301,log a 3 = 0.477andlog a 5 = 0.699
evaluate
a
log a 20
___________________________________________________
___________________________________________________
___________________________________________________
b
log a 24
___________________________________________________
Part 3
Logarithms
59
___________________________________________________
___________________________________________________
c
log a 45
___________________________________________________
___________________________________________________
___________________________________________________
d
log a 50
___________________________________________________
___________________________________________________
___________________________________________________
e
log a 2
1
4
___________________________________________________
___________________________________________________
___________________________________________________
f
log a
24
25
___________________________________________________
___________________________________________________
___________________________________________________
g
log a 6.25
___________________________________________________
___________________________________________________
___________________________________________________
60
PAS5.3.8 Functions and Logarithms
3
Evaluate
a
1.5 log 3 9
___________________________________________________
___________________________________________________
___________________________________________________
b
4
log 5 125
3
___________________________________________________
___________________________________________________
c
___________________________________________________
2
log 2 32
5
___________________________________________________
___________________________________________________
d
___________________________________________________
3
log 3 81
4
___________________________________________________
___________________________________________________
e
___________________________________________________
3
1
log 3
2
9
___________________________________________________
___________________________________________________
f
___________________________________________________
2
1
log 4
3
64
___________________________________________________
___________________________________________________
___________________________________________________
Part 3
Logarithms
61
Exercise 3.3 – Using logarithm laws
1
Simplify these by writing as a single logarithm.
a
2 log x − 3log y
___________________________________________________
___________________________________________________
___________________________________________________
b
___________________________________________________
1
log x + log y − 2 log z
2
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
c
log 4 − 4 log 5 + 3log 50
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
d
___________________________________________________
1
2 log 3 − log 2 + log 9
2
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
62
PAS5.3.8 Functions and Logarithms
2
Write these in expanded form.
a
log 2
4 p2
q
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b
log x
x2
y
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
3
Solve, giving answers correct to 3 decimal places.
a
5 = 3x
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b
2 m = 11
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
c
2 2 k +1 = 7
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Part 3
Logarithms
63
d
5 y = 3y + 3
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
e
6 2 x −1 = 3x
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
4
Solve 4 t +1 =
1
8 2
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
5
Solve log 27 3 = x
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
64
PAS5.3.8 Functions and Logarithms
Exercise 3.4 – Exponentials and logarithms
1
Complete the table of values for y = log 2 x .
a
x
y
b
–3
–2
–1
0
1
2
3
1
Draw the graph of y = log 2 xfor ≤ x ≤ 8 .
8
y
4
3
2
1
–1
1
0
2
3
4
5
6
7
8
x
–1
–2
–3
–4
2
Two curves y = a x andy = log a x are shown.
y
7
6
5
4
3
2
1
–1
0
1
2
3
4
5
6
7
x
–1
–2
–3
a
What is the value of a? ________________________________
How did you work it out? Explain.
___________________________________________________
___________________________________________________
Part 3
Logarithms
65
___________________________________________________
b
On this graph draw in the line y = x.
c
What is the important link between y = a x andy = log a x , and
this line?
___________________________________________________
___________________________________________________
3
For the diagram below, find
a
the co-ordinates of Q _________________________________
b
the area of the rectangle OPQR.
___________________________________________________
___________________________________________________
y
g2 x
lo
y=
P
Q
R
0
4
x
4
In the figure below, NM is drawn parallel to OX. Find the length of
MN.
y=
2x
y
N
y=
log 2
x
Q
R
0
4
x
_______________________________________________________
_______________________________________________________
_______________________________________________________
66
PAS5.3.8 Functions and Logarithms
Find the co-ordinates of A and B, and hence calculate the length of
AB.
2x
5
y=
y
4
0
A
B
x=
2y
x
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Part 3
Logarithms
67
Exercise 3.5 – Miscellaneous exercises
1
If log a 2 = pandlog a 3 = q , give the values of the following in
terms of p and/or q.
a
log a 32
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b
log a 81
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c
log a 18
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d
log a 72
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2
If log 5 a = p , what is the value of
a
log 5 25a
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PAS5.3.8 Functions and Logarithms
b
log 5 5a 2
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3
Express y in terms of x without the use of logs.
a
log 5x – log 2y = log 8
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b
log 6x + log y = 2
[Hint: log 100 = 2]
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c
2log x – 3log y = log 9
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Part 3
Logarithms
69
4
Solve the following equations.
a
log 3x + log 2 = log (x + 1)
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b
log (x + 3) – log (x – 1) = log 4
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c
log y + log (y – 1) = log (5 – y)
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d
log (z + 5) – log 3z = log (z + 1)
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PAS5.3.8 Functions and Logarithms
5
Simplify
a
log 81 ÷ log 27
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b
log 16 ÷ log 128
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c
log 4 ÷ log
1
4
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d
log a 4 ÷ log a
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e
log m 3 ÷ log 3 m
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f
log b ÷ log
1
b
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Part 3
Logarithms
71
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The following is part of the graph for the function y = log 7 x .
6
y
0.5
0
0.5
1
1.5
2
2.5
3x
–0.5
Use the graph to answer the following questions, to an accuracy
allowed by the graph.
a
Find
i
log 7 2
ii
log 7 1.4
iii
log 7 0.6
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b
Find the value of x for which
i
log 7 x = 0
ii
log 7 x = 0.5
iii
log 7 x = −0.7
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7
8
True or false.
a
The log of a number between 0 and 1 is negative.____________
b
The log of a negative number is less than zero. ______________
c
Double the number and its logarithm is doubled too. _________
d
The line x = 0 is an asymptote to y = log 7 x .________________
What is the equation of the inverse function to y = log 7 x ?
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PAS5.3.8 Functions and Logarithms